Generating random numbers
|
|
- Adele Powell
- 7 years ago
- Views:
Transcription
1 Generating random numbers Lecturer: Dmitri A. Moltchanov
2 OUTLINE: Why do we need random numbers; Basic steps in generation; Uniformly distributed random numbers; Statistical tests for uniform random numbers; Random numbers with arbitrary distributions; Statistical tests for random numbers with arbitrary distribution; Multidimensional distributions. Lecture: Generating random numbers 2
3 1. The need for random numbers Examples of randomness in telecommunications: interarrival times between arrivals of packets, tasks, etc.; service time of packets, tasks, etc.; time between failure of various components; repair time of various components;... Importance for simulations: random events are characterized by distributions; simulations: we cannot use distribution directly. For example, M/M/1 queuing system: arrival process: exponential distribution with mean 1/λ; service times: exponential distribution with mean 1/µ. Lecture: Generating random numbers 3
4 Discrete-event simulation of M/M/1 queue INITIALIZATION time:=0; queue:=0; sum:=0; throughput:=0; generate first interarrival time; MAIN PROGRAM while time < runlength do case nextevent of arrival event: time:=arrivaltime; add customer to a queue; start new service if the service is idle; generate next interarrival time; departure event: time:=departuretime; throughput:=throughtput + 1; remove customer from a queue; if (queue not empty) sum:=sum + waiting time; start new service; OUTPUT mean waiting time = sum / throughput Lecture: Generating random numbers 4
5 2. General notes General approach nowadays: transforming one random variable to another one; as a reference distribution a uniform distribution is often used. Note the following: most simulators contain generator of uniformly distributed numbers in interval (0, 1). may not contain arbitrarily distributed random numbers you want. The procedure is to: generate RN with inform distribution between a and b, b >>>> a; transform it somehow to random number with uniform distribution on (0, 1); transform it somehow to a random number with desired distribution. Lecture: Generating random numbers 5
6 2.1. Pseudo random numbers All computer generated numbers are pseudo ones: we know the method how they are generated; we can predict any random sequence in advance. The goal is then: imitate random sequences as good as possible. Requirements for generators: must be fast; must have low complexity; must have sufficiently long cycles; must allow to generate repeatable sequences; must be independent; must closely follow a given distribution. Lecture: Generating random numbers 6
7 2.2. Step 1: uniform random numbers in (a, b) Basic approach: generate random number with uniform distribution on (a, b); transform these random numbers to (0, 1); transform it somehow to a random number with desired distribution. Uniform generators: old methods: mostly based on radioactivity; Von Neumann s algorithm; congruential methods. Basic approach: next number is some function of previous one γ i+1 = F (γ i ), i = 0, 1,..., (1) recurrence relation of the first order; γ 0 is known and directly computed from the seed. Lecture: Generating random numbers 7
8 2.3. Step 2: transforming to random numbers in (0, 1) Basic approach: generate random number with uniform distribution on (0, 1); transform these random numbers to (0, 1); transform it somehow to a random number with desired distribution. Uniform U(0, 1) distribution has the following pdf: 1, 0 x 1 f(x) = 0, otherwise. (2) Lecture: Generating random numbers 8
9 Mean and variance are given by: E[X] = 1 0 xdx = x = 1 2, σ 2 [X] = (3) How to get U(0, 1): by rescaling from U(0, m) as follows: y i = γ i /m, (4) where m is the modulo in linear congruential algorithm. What we get: something like: 0.12, 0.67, 0.94, 0.04, 0.65, 0.20,... ; sequence that appears to be random... Lecture: Generating random numbers 9
10 2.4. Step 3: non-uniform random numbers Basic approach: generate random number with uniform distribution on (a, b); transform these random numbers to (0, 1); transform it somehow to a random number with desired distribution. If we have generator U(0, 1) the following techniques are avalable: discretization: bernoulli, binomial, poisson, geometric; rescaling: uniform; inverse transform: exponential; specific transforms: normal; rejection method: universal method; reduction method: Erlang, Binomial; composition method: for complex distributions. Lecture: Generating random numbers 10
11 3. Uniformly distributed random numbers The generator is fully characterized by (S, s 0, f, U, g): S is a finite set of states; s 0 S is the initial state; f(s S) is the transition function; U is a finite set of output values; g(s U) is the output function. The algorithm is then: let u 0 = g(s 0 ); for i = 1, 2,... do the following recursion: s i = f(s i 1 ); u i = g(s i ). Note: functions f( ) and g( ) influence the goodness of the algorithm heavily. Lecture: Generating random numbers 11
12 user choice s 0 u 0 =g(s 0 ) u 0 u 4 s 1 =f(s 0 ) s 1 s 0 s 3 s 4 s 4 =f(s 3 ) u 1 =g(s 1 ) u 4 =g(s 4 ) u 3 =g(s 3 ) u 1 u 3 s 2 =f(s 1 ) s 3 =f(s 2 ) s 2 u 2 u 2 =g(s 2 ) Figure 1: Example of the operations of random number generator. Here s 0 is a random seed: allows to repeat the whole sequence; allows to manually assure that you get different sequence. Lecture: Generating random numbers 12
13 3.1. Von Neumann s generator The basic procedure: start with some number u 0 of a certain length x (say, x = 4 digits, this is seed); square the number; take middle 4 digits to get u 1 ; repeat... example: with seed 1234 we get 1234, 5227, 3215, 3362, 3030, etc. Shortcoming: sensitive to the random seed: seed 2345: 2345, 4990, 9001, 180, 324, 1049, 1004, 80, 64, (will always < 100); may have very short period: seed 2100: 2100, 4100, 8100, 6100, 2100, 4100, 8100,... (period = 4 numbers). To generate U(0, 1): divide each obtained number by 10 x (x is the length of u 0 ). Note: this generator is also known as midsquare generator. Lecture: Generating random numbers 13
14 3.2. Congruential methods There are a number of versions: additive congruential method; multiplicative congruential method; linear congruential method; Tausworthe binary generator. General congruential generator: u i+1 = f(u i, u i 1,... ) mod m, (5) u i, u i 1,... are past numbers. For example, quadratic congruential generator: u i+1 = (a 1 u 2 i + a 2 u i 1 + c) mod m. (6) Note: if here a 1 = a 2 = 1, c = 0, m = 2 we have the same as midsquare method. Lecture: Generating random numbers 14
15 3.3. Additive congruential method Additive congruential generator is given: u i+1 = (a 1 u i + a 2 u i a k u i k ) mod m. (7) The common special case is sometimes used: u i+1 = (a 1 u i + a 2 u i 1 ) mod m. (8) Characteristics: divide by m to get U(0, 1); maximum period is m k ; note: rarely used. Shortcomings: consider k = 2: consider three consecutive numbers u i 2, u i 1, u i ; we will never get: u i 2 < u i < u i 1 and u i 1 < u i < u i 2 (must be 1/6 of all sequences). Lecture: Generating random numbers 15
16 3.4. Multiplicative congruential method Multiplicative congruential generator is given: u i+1 = (au i ) mod m. (9) Characteristics: divide by m to get U(0, 1); theoretical maximum period is m; note: rarely used. Shortcomings: can never produce 0. Choice of a, m is very important: recommended m = (2 p 1) with p = 2, 3, 5, 7, 13, 17, 19, 31, 61 (Fermat numbers); if m = 2 q, q 4 simplifies the calculation of modulo; practical maximum period is at best no longer than m/4. Lecture: Generating random numbers 16
17 3.5. Linear congruential method Linear congruential generator is given: u i+1 = (au i + c) mod m, (10) where a, c, m are all positive. Characteristics: divide by m to get U(0, 1); maximum period is m; frequently used. Choice of a, c, m is very important. To get full period m choose: m and c have no common divisor; c and m are prime number (distinct natural number divisors 1 and itself only); if q is a prime divisor of m then a = 1, mod q; if 4 is a divisor of m then a = 1, mod 4. Lecture: Generating random numbers 17
18 The step-by-step procedure is as follows: set the seed x 0 ; multiply x by a and add c; divide the result by m; the reminder is x 1 ; repeat to get x 2, x 3,.... Examples: x 0 = 7, a = 7, c = 7, m = 10 we get: 7,6,9,0,7,6,9,0,... (period = 4); x 0 = 1, a = 1, c = 5, m = 13 we get: 1,6,11,3,8,0,5,10,2,7,12,4,9,1... (period = 13); x 0 = 8, a = 2, c = 5, m = 13 we get: 8,8,8,8,8,8,8,8,... (period = 1!). Recommended values: a = 314, 159, 269, c = 453, 806, 245, m = 231 for 32 bit machine. Lecture: Generating random numbers 18
19 Complexity of the algorithm: addition, multiplications and division: division is slow: to avoid it set m to the size of the computer word. Overflow problem when m equals to the size of the word: values a, c and m are such that the result ax i + c is greater than the word; it may lead to loss of significant digits but it does not hurt! How to deal with: register can accommodate 2 digits at maximum; the largest number that can be stored is 99; if m = 100: for a = 8, u 0 = 2, c = 10 we get (au i + c) mod 100 = 26; if m = 100: for a = 8, u 0 = 20, c = 10 we get (au i + c) mod 100 = 170; au i = 8 20 = 160 causing overflow; first significant digit is lost and register contains 60; the reminder in the register (result) is: ( ) mod 70 = 70. the same as 170 mod 100 = 70. Lecture: Generating random numbers 19
20 3.6. How to get good congruental generator Characteristics of good generator: should provide maximum density: no large gaps in [0, 1] are produced by random numbers; problem: each number is discrete; solution: a very large integer for modulus m. should provide maximum period: achieve maximum density and avoid cycling; achieve by: proper choice of a, c, m, and x 0. effective for modern computers: set modulo to power of 2. Lecture: Generating random numbers 20
21 3.7. Tausworthe generator Tausworthe generator (case of linear congruential generator or order k): ( k ) z i = (a 1 z i 1 + a 2 z i a k z i k + c) mod 2 = a j z i j + c mod 2. (11) where a j {0, 1}, j = 0, 1,..., k; the output is binary: j=1 Advantages: independent of the system (computer architecture); independent of the word size; very large periods; can be used in composite generators (we consider in what follows). Note: there are several bit selection techniques to get numbers. Lecture: Generating random numbers 21
22 A way to generate numbers: choose an integer l k; split in blocks of length l and interpret each block as a digit: u n = l 1 j=0 z nl+j 2 (j+1). (12) In practice, only two a i are used and set to 1 at places h and k. We get: Example: h = 3, k = 4, initial values 1,1,1,1; we get: ; period is 2 k 1 = 15; if l = 4: 13/16, 7/16, 8/16, 9/16, 10/16, 15/16, 1/16, 3/16... z n = (z i h + z i k ) mod 2. (13) Lecture: Generating random numbers 22
23 3.8. Composite generator Idea: use two generators of low period to generate another with wider period. The basic principle: use the first generator to fill the shuffling table (address - entry (random number)); use random numbers of second generator as addresses in the next step; each number corresponding to the address is replaced by new random number of first generator. The following algorithm uses one generator to shuffle with itself: 1. create shuffling table of 100 entries (i, t i = γ i, i = 1, 2,..., 100); 2. draw random number γ k and normalize to the range (1, 100); 3. entry i of the table gives random number t i ; 4. draw the next random number γ k+1 and update t i = γ k+1 ; 5. repeat from step 2. Note: table with 100 entries gives fairly good results. Lecture: Generating random numbers 23
24 4. Tests for random number generators What do we want to check: independence; uniformity. Important notes: if and only if tests passed number can be treated as random; recall: numbers are actually deterministic! Commonly used tests for independence: runs test; correlation test. Commonly used tests for uniformity: Kolmogorov s test; χ 2 test. Lecture: Generating random numbers 24
25 4.1. Independence: runs test Basic idea: compute patterns of numbers (always increase, always decrease, etc.); compare to theoretical probabilities. 1/3 1/3 1/3 1/3 1/3 1/3 Figure 2: Illustration of the basic idea. Lecture: Generating random numbers 25
26 Do the following: consider a sequence of pseudo random numbers: {u i, i = 0, 1,..., n}; consider unbroken subsequences of numbers where numbers are monotonically increasing; such subsequence is called run-up; example: 0.78,081,0.89,0.81 is a run-up of length 3. compute all run-ups of length i: r i, i = 1, 2, 3, 4, 5; all run-ups of length i 6 are grouped into r 6. calculate: R = 1 n 1 i,j 6 (r i nb i )(r j nb j )a ij, 1 i, j 6, (14) where (b 1, b 2,..., b 6 ) = ( 1 6, 5 24, , 19 ) 720, , 1, (15) 840 Lecture: Generating random numbers 26
27 Coefficients a ij must be chosen as an element of the matrix: Statistics R has χ 2 distribution: number of freedoms: 6; n > If so, observations are i.i.d. Lecture: Generating random numbers 27
28 4.2. Independence: correlation test Basic idea: compute autocorrelation coefficient for lag-1; if it is not zero and this is statistically significant result, numbers are not independent. Compute statistics (lag-1 autocorrelation coefficient) as: R = N (u j E[u])(u j+1 E[u])/ j=1 N (u j E[j]) 2. (16) j=1 Practice: if R is relatively big there is serial correlation. Important notes: exact distribution of R is unknown; for large N: if u j uncorrelated we have: P r{ 2/ N R 2/ N}; therefore: reject hypotheses of non-correlated at 5% level if R is not in { 2/ N, 2/ N}. Notes: other tests for correlation Ljung and Box test, Portmanteau test, etc. Lecture: Generating random numbers 28
29 4.3. Uniformity: χ 2 test The algorithm: divide [0, 1] into k, k > 100 non-overlapping intervals; compute the relative frequencies of falling in each category, f i : ensure that there are enough numbers to get f i > 5, i = 1, 2,..., k; values f i > 5, i = 1, 2,..., k are called observed values. if observations are truly uniformly distributed then: these values should be equal to r i = n/k, i = 1, 2,..., k; these values are called theoretical values. compute χ 2 statistics for uniform distribution: χ 2 = k n that must have k 1 degrees of freedom. k i=1 ( f i n k ) 2. (17) Lecture: Generating random numbers 29
30 Hypotheses: H 0 observations are uniformly distributed; H 1 observations are not uniformly distributed. H 0 is rejected if: computed value of χ 2 is greater than one obtained from the tables; you should check the entry with k 1 degrees of freedom and 1-a level of significance. Lecture: Generating random numbers 30
31 4.4. Kolmogorov test Facts about this test: compares empirical distribution with theoretical ones; empirical: F N (x) number of smaller than or equal to x, divided by N; theoretical: uniform distribution in (0, 1): F (x) = x, 0 < x < 1. Hypotheses: H 0 : F N (x) follows F (x); H 1 : F N (x) does not follow F (x). Statistics: maximum absolute difference over a range: R = max F (x) F N (x). (18) if R > R α : H 0 is rejected; if R R α : H 0 is accepted. Note: use tables for N, α (significance level), to find R α. Lecture: Generating random numbers 31
32 Example: we got 0.44, 0.81, 0.14, 0.05, 0.93: H 0 : random numbers follows uniform distribution; we have to compute: R (j) j/n j/n R (j) R (j) (j-1)/n compute statistics as: R = max F (x) F N (x) = 0.26; from tables: for α = 0.05, R α = > R; H 0 is accepted, random numbers are distributed uniformly in (0, 1). Lecture: Generating random numbers 32
33 4.5. Other tests The serial test: consider pairs (u 1, u 2 ), (u 3, u 4 ),..., (u 2N 1, u 2N ); count how many observations fall into N 2 different subsquares of the unit square; apply χ 2 test to decide whether they follow uniform distribution; one can formulate M-dimensional version of this test. The permutation test look at k-tuples: (u 1, u k ), (u k+1, u 2k ),..., (u (N 1)k+1, u Nk ); in a k-tuple there k! possible orderings; in a k-tuple all orderings are equally likely; determine frequencies of orderings in k-tuples; apply χ 2 test to decide whether they follow uniform distribution. Lecture: Generating random numbers 33
34 The gap test let J be some fixed subinterval in (0, 1); if we have that: u n+j not in J, 0 j k, and both u n 1 J, u n+k+1 J; we say that there is a gap of length k. H 0 : numbers are independent and uniformly distributed in (0, 1): gap length must be geometrically distributed with some parameter p; p is the length of interval J: P r{gap of length k} = p(1 p) k. (19) practice: we observe a large number of gaps, say N; choose an integer and count number of gaps of length 0, 1,..., h 1 and h; apply χ 2 test to decide whether they independent and follow uniform distribution. Lecture: Generating random numbers 34
35 4.6. Important notes Some important notes on seed number: do not use seed 0; avoid even values; do not use the same sequence for different purposes in a single simulation run. Note: these instruction may not be applicable for a particular generator. General notes: some common generators are found to be inadequate; even if generator passed tests, some underlying pattern might still be undetected; if the task is important use composite generator. Lecture: Generating random numbers 35
36 5. Random numbers with arbitrary distribution Discrete distributions: discretization; for any discrete distribution. rescaling: for uniform random numbers in (a, b). methods for specific distributions. Continuous distributions: inverse transform; rejection method; composition method; methods for specific distributions. Lecture: Generating random numbers 36
37 5.1. Discrete distributions: discretization Consider arbitrary distributed discrete RV: P r{x = x j } = p j, j = 0, 1,..., p j = 1. (20) j=0 The following method can be applied: generate uniformly distributed RV; use the following to generate discrete RV: this method can be applied to any discrete RV; there are some specific methods for specific discrete RVs. Lecture: Generating random numbers 37
38 Figure 3: Illustration of the proposed approach. Lecture: Generating random numbers 38
39 The step-by-step procedure: compute probabilities p i = P r{x = x i }, i = 0, 1,... ; generate RV u with U(0, 1); if u < p 0, set X = x 0 ; if u < p 0 + p 1, set X = x 1 ; if u < p 0 + p 1 + p 2, set X = x 2 ;... Note the following: this is inverse transform method for discrete RVs: we determine the value of u; we determine the interval [F (x i 1 ), F (x i )] in which it lies. complexity depends on the number of intervals to be searched. Lecture: Generating random numbers 39
40 Example: p 1 = 0.2, p 2 = 0.1, p 3 = 0.25, p 4 = 0.45: determine generator for P r{x = x j } = p j. Algorithm 1: generate u = U(0, 1); if u < 0.2, set X = 1, return; if u < 0.3, set X = 2; if u < 0.55, set X = 3; set X = 4. Algorithm 2 (more effective): generate u = U(0, 1); If u < 0.45, set X = 4; if u < 0.7, set X = 3; if u < 0.9, set X = 1; set X = 2. Lecture: Generating random numbers 40
41 5.2. Example of discretization: Poisson RV Poisson RV have the following distribution: p i = P r{x = i} = λi e λ, i! i = 0, 1,.... (21) We use the property: p i+1 = λ i + 1 p i, i = 1, 2,.... (22) The algorithm: 1. generate u = U(0, 1); 2. i = 0, p = e λ, F = p; 3. if u < F, set X = i; 4. p = λp/(i + 1), F = F + p, i = i + 1; 5. go to step 3. Lecture: Generating random numbers 41
42 5.3. Example of discretization: binomial RV Binomial RV have the following distribution: p i = P r{x = i} = We are going to use the following property: The algorithm: 1. generate u = U(0, 1); p i+1 = n i i c = p/(1 p), i = 0, d = (1 p)n, F = d; 3. if u < F, set X = i 4. d = [c(n i)/(i + 1)]d, F = F + d, i = i + 1; 5. go to step 3. n! i!(n i)! pi (1 p) n i, i = 0, 1,.... (23) p 1 p p i, i = 0, 1,.... (24) Lecture: Generating random numbers 42
43 5.4. Continuous distributions: inverse transform method Inverse transform method: applicable only when cdf can be inversed analytically; works for a number of distributions: exponential, unform, Weibull, etc. Assume: we would like to generate numbers with pdf f(x) and cdf F (x); recall, F (x) is defined on [0, 1]. The generic algorithm: generate u = U(0, 1); set F (x) = u; find x = F 1 (u), F 1 ( ) is the inverse transformation of F ( ). Lecture: Generating random numbers 43
44 Example: we want to generate numbers from the following pdf f(x) = 2x, 0 x 1; calculate the cdf as follows: F (x) = x 0 2tdt = x 2, 0 x 1. (25) let u be the random number, we have u = x 2 or u = x; get the random number. Lecture: Generating random numbers 44
45 5.5. Inverse transform method: uniform continuous distribution Uniform continuous distribution has the following pdf and cdf: 1 (b a) f(x) =, a < x < b (x a) 0, otherwise, F (x) =, a < x < b (b a) 0, otherwise. (26) The algorithm: generate u = U(0, 1); set u = F (x) = (x a)/(b a); solve to get x = a + (b a)u. Lecture: Generating random numbers 45
46 5.6. Inverse transform method: exponential distribution Exponential distribution has the following pdf and cdf: f(x) = λe λx, F (x) = 1 e λx, λ > 0, x 0. (27) The algorithm: generate u = U(0, 1); set u = F (x) = e λx ; solve to get x = (1/λ) log u. Lecture: Generating random numbers 46
47 5.7. Inverse transform method: Erlang distribution Erlang distribution: convolution of k exponential distributions. The algorithm: generate u = U(0, 1); sum of exponential variables x 1,..., x k with mean 1/λ; solve to get: x = k x i = 1 λ i=1 k log u i = 1 k λ log u i. (28) i=1 i=1 Lecture: Generating random numbers 47
48 5.8. Specific methods: normal distribution Normal distribution has the following pdf: f(x) = 1 σ 1 2π e 2 (x µ) 2 where σ and µ are the standard deviation and the mean. σ 2, < x <, (29) Standard normal distribution (RV Z = (X µ/)σ) has the following pdf: f(z) = 1 2π e 1 2 z2, < z <, where µ = 0, σ = 1. (30) Lecture: Generating random numbers 48
49 Central limit theorem: if x 1, x 2,..., x n are independent with E[x i ] = µ, σ 2 [x i ] = σ 2, i = 1, 2,..., n; the sum of them approaches normal distribution if n : E[ x i ] = nµ, σ 2 [ x i ] = nσ 2. The approach: generate k random numbers u i = U(0, 1), i = 0, 1,..., k 1; each random numbers has: E[u i ] = (0 + 1)/2 = 1/2, σ 2 [u i ] = (1 0) 2 /12 = 1/12; sum of these number follows normal distribution with: ( ) k ui N 2, k ui k/2, or 12 k/ N(0, 1). (31) 12 if the RV we want to generate is x with mean µ and standard deviation σ: x µ σ finally (note that k should be at least 10): x µ ui k/2 12 = σ k/, or x = σ 12 k N(0, 1). (32) ( ui k ) + µ. (33) 2 Lecture: Generating random numbers 49
50 5.9. Specific method: empirical continuous distributions Assume we have a histogram: x i is the midpoint of the interval i; f(x i ) is the length of the ith rectangle. Note: the task is different from sampling from discrete distribution. Lecture: Generating random numbers 50
51 Construct the cdf as follows: F (x i ) = k {F (x i 1 ),F (x i )} f(x k ), (34) which is monotonically increasing within each interval [F (x i 1 ), F (x i )]. Lecture: Generating random numbers 51
52 The algorithm: generate u = U(0, 1); assume that u {F (x i 1 ), F (x i )}; use the following linear interpolation to get: u F (x i 1 ) x = x i 1 + (x i x i 1 ) F (x i ) F (x i 1 ). (35) Note: this approach can also be used for analytical continuous distribution. get (x i, f(x i )), i = 1, 2,..., k and follow the procedure. Lecture: Generating random numbers 52
53 5.10. Rejection method Works when: pdf f(x) is bounded; x has a finite range, say a x b. The basic steps are: normalize the range of f(x) by a scale factor such that cf(x) 1, a x b; define x as a linear function of u = U(0, 1), i.e. x = a + (b a)u; generate pairs of random numbers (u 1, u 2 ), u 1, u 2 = U(0, 1); accept the pair and use x = a + (b a)u 1 whenever: the pair satisfies u 2 cf(a + (b a)u 1 ); meaning that the pair (x, u 2 ) falls under the curve of cf(x). Lecture: Generating random numbers 53
54 The underlying idea: P r{u 2 cf(x)} = cf(x); if x is chosen at random from (a, b): we reject if u 2 > cf(x); we accept if u 2 cf(x); we match f(x). Lecture: Generating random numbers 54
55 Example: generate numbers from f(x) = 2x, 0 x 1: 1. select c such that cf(x) 1: for example: c = generate u 1 and set x = u 1 ; 3. generate u 2 : if u 2 < cf(u 1 ) = (0.5)2u 1 = u 1 then accept x; otherwise go back to step 2. Lecture: Generating random numbers 55
56 5.11. Convolution method The basis of the method is the representation of cdf F (x): F (x) = p j F j (x), (36) j=1 p j 0, j = 1, 2,..., j=1 p j = 1. Works when: it is easy to to generate RVs with distribution F j (x) than F (x); hyperexponential RV; Erlang RV. The algorithm: 1. generate discrete RV J, P r{j = j} = p j ; 2. given J = j generate RV with F j (x); 3. compute j=1 p jf j (x). Lecture: Generating random numbers 56
57 Example: generate from exponential distribution: divide (0, ) into intervals (i, i + 1), i = 0, 1,... ; the probabilities of intervals are given: p i = P r{i X < i + 1} = e i e (i+1) = e i (1 e 1 ), (37) gives geometric distribution. the conditional pdfs are fiven by: f i (x) = e (x i) /(1 e 1 ), i x < i + 1. (38) in the interval i(x i) has the pdf e x /(1 e 1 ), 0 x < 1. The algorithm: get I from geometric distribution p i = e i /(1 e 1 ), i = 0, 1,... ; get Y from e x /(1 e 1 ), 0 x < 1; X = I + Y. Lecture: Generating random numbers 57
58 6. Statistical tests for RNs with arbitrary distribution What we have to test for: independence; particular distribution. Tests for independence: correlation tests: Portmanteau test, modified Portmanteau test, ±2/ n, etc. note: here we test only for linear dependence... Tests for distribution: χ 2 test; Kolmogorov s test. Lecture: Generating random numbers 58
59 7. Multi-dimensional distributions Task: generate samples from RV (X 1, X 2,..., X n ). Write the joint density function as: f(x 1, x 2,..., x n ) = f 1 (x 1 )f 2 (x 2 x 1 )... f(x n x 1... x n 1 ). (39) f 1 (x 1 ) is the marginal distribution of X 1 ; f k (x k x 1,..., x k 1 ) is the conditional pdf of X k with condition on X 1 = x 1,..., X k 1 = x k 1. The basic idea: generate one number at a time: get x 1 from f 1 (x 1 ); get x 2 from f 2 (x 2 x 1 ), etc. The algorithm: get n random numbers u i = U(0, 1), i = 0, 1,..., n; subsequently get the following RVs: F 1 (X 1 ) = u 1, F 2 (X 2 X 1 ) = u 2,... F n (X n X 1,..., X n 1 ) = u n. (40) Lecture: Generating random numbers 59
60 Example: generate from f(x, y) = x + y: marginal pdf and cdf of X are given by: f(x) = 1 0 f(x, y)dy = x + 1 2, x F (x) = f(x )dx = 1 2 (x2 + x). (41) 0 conditional pdf and cdf of Y are given by: f(y x) = f(x, y) f(x) = x + y x + 1, F (y x) = 2 y 0 f(y x)dy = xy y2 x + 1. (42) 2 by inversion we get: x = 1 2 ( 8u ), y = x 2 + u 2 (1 + 2x) x. (43) Lecture: Generating random numbers 60
2WB05 Simulation Lecture 8: Generating random variables
2WB05 Simulation Lecture 8: Generating random variables Marko Boon http://www.win.tue.nl/courses/2wb05 January 7, 2013 Outline 2/36 1. How do we generate random variables? 2. Fitting distributions Generating
More informationProbability density function : An arbitrary continuous random variable X is similarly described by its probability density function f x = f X
Week 6 notes : Continuous random variables and their probability densities WEEK 6 page 1 uniform, normal, gamma, exponential,chi-squared distributions, normal approx'n to the binomial Uniform [,1] random
More informationTesting Random- Number Generators
Testing Random- Number Generators Raj Jain Washington University Saint Louis, MO 63130 Jain@cse.wustl.edu Audio/Video recordings of this lecture are available at: http://www.cse.wustl.edu/~jain/cse574-08/
More informationChapter 3 RANDOM VARIATE GENERATION
Chapter 3 RANDOM VARIATE GENERATION In order to do a Monte Carlo simulation either by hand or by computer, techniques must be developed for generating values of random variables having known distributions.
More informationProbability and Statistics Prof. Dr. Somesh Kumar Department of Mathematics Indian Institute of Technology, Kharagpur
Probability and Statistics Prof. Dr. Somesh Kumar Department of Mathematics Indian Institute of Technology, Kharagpur Module No. #01 Lecture No. #15 Special Distributions-VI Today, I am going to introduce
More informationLecture 13 - Basic Number Theory.
Lecture 13 - Basic Number Theory. Boaz Barak March 22, 2010 Divisibility and primes Unless mentioned otherwise throughout this lecture all numbers are non-negative integers. We say that A divides B, denoted
More informationNotes on Continuous Random Variables
Notes on Continuous Random Variables Continuous random variables are random quantities that are measured on a continuous scale. They can usually take on any value over some interval, which distinguishes
More informationIntroduction to Probability
Introduction to Probability EE 179, Lecture 15, Handout #24 Probability theory gives a mathematical characterization for experiments with random outcomes. coin toss life of lightbulb binary data sequence
More informationStatistics 100A Homework 7 Solutions
Chapter 6 Statistics A Homework 7 Solutions Ryan Rosario. A television store owner figures that 45 percent of the customers entering his store will purchase an ordinary television set, 5 percent will purchase
More informationProbability and Random Variables. Generation of random variables (r.v.)
Probability and Random Variables Method for generating random variables with a specified probability distribution function. Gaussian And Markov Processes Characterization of Stationary Random Process Linearly
More informationRandom Variate Generation (Part 3)
Random Variate Generation (Part 3) Dr.Çağatay ÜNDEĞER Öğretim Görevlisi Bilkent Üniversitesi Bilgisayar Mühendisliği Bölümü &... e-mail : cagatay@undeger.com cagatay@cs.bilkent.edu.tr Bilgisayar Mühendisliği
More informationOverview of Monte Carlo Simulation, Probability Review and Introduction to Matlab
Monte Carlo Simulation: IEOR E4703 Fall 2004 c 2004 by Martin Haugh Overview of Monte Carlo Simulation, Probability Review and Introduction to Matlab 1 Overview of Monte Carlo Simulation 1.1 Why use simulation?
More informationInstitute of Actuaries of India Subject CT3 Probability and Mathematical Statistics
Institute of Actuaries of India Subject CT3 Probability and Mathematical Statistics For 2015 Examinations Aim The aim of the Probability and Mathematical Statistics subject is to provide a grounding in
More informationLecture 7: Continuous Random Variables
Lecture 7: Continuous Random Variables 21 September 2005 1 Our First Continuous Random Variable The back of the lecture hall is roughly 10 meters across. Suppose it were exactly 10 meters, and consider
More informationSums of Independent Random Variables
Chapter 7 Sums of Independent Random Variables 7.1 Sums of Discrete Random Variables In this chapter we turn to the important question of determining the distribution of a sum of independent random variables
More informationAlgebra 2 Chapter 1 Vocabulary. identity - A statement that equates two equivalent expressions.
Chapter 1 Vocabulary identity - A statement that equates two equivalent expressions. verbal model- A word equation that represents a real-life problem. algebraic expression - An expression with variables.
More informationFEGYVERNEKI SÁNDOR, PROBABILITY THEORY AND MATHEmATICAL
FEGYVERNEKI SÁNDOR, PROBABILITY THEORY AND MATHEmATICAL STATIsTICs 4 IV. RANDOm VECTORs 1. JOINTLY DIsTRIBUTED RANDOm VARIABLEs If are two rom variables defined on the same sample space we define the joint
More informationMultiple Choice: 2 points each
MID TERM MSF 503 Modeling 1 Name: Answers go here! NEATNESS COUNTS!!! Multiple Choice: 2 points each 1. In Excel, the VLOOKUP function does what? Searches the first row of a range of cells, and then returns
More informationJitter Measurements in Serial Data Signals
Jitter Measurements in Serial Data Signals Michael Schnecker, Product Manager LeCroy Corporation Introduction The increasing speed of serial data transmission systems places greater importance on measuring
More informationImportant Probability Distributions OPRE 6301
Important Probability Distributions OPRE 6301 Important Distributions... Certain probability distributions occur with such regularity in real-life applications that they have been given their own names.
More informationLECTURE 16. Readings: Section 5.1. Lecture outline. Random processes Definition of the Bernoulli process Basic properties of the Bernoulli process
LECTURE 16 Readings: Section 5.1 Lecture outline Random processes Definition of the Bernoulli process Basic properties of the Bernoulli process Number of successes Distribution of interarrival times The
More informationCHAPTER 6: Continuous Uniform Distribution: 6.1. Definition: The density function of the continuous random variable X on the interval [A, B] is.
Some Continuous Probability Distributions CHAPTER 6: Continuous Uniform Distribution: 6. Definition: The density function of the continuous random variable X on the interval [A, B] is B A A x B f(x; A,
More informationAachen Summer Simulation Seminar 2014
Aachen Summer Simulation Seminar 2014 Lecture 07 Input Modelling + Experimentation + Output Analysis Peer-Olaf Siebers pos@cs.nott.ac.uk Motivation 1. Input modelling Improve the understanding about how
More informationIEOR 6711: Stochastic Models I Fall 2012, Professor Whitt, Tuesday, September 11 Normal Approximations and the Central Limit Theorem
IEOR 6711: Stochastic Models I Fall 2012, Professor Whitt, Tuesday, September 11 Normal Approximations and the Central Limit Theorem Time on my hands: Coin tosses. Problem Formulation: Suppose that I have
More informationECE 842 Report Implementation of Elliptic Curve Cryptography
ECE 842 Report Implementation of Elliptic Curve Cryptography Wei-Yang Lin December 15, 2004 Abstract The aim of this report is to illustrate the issues in implementing a practical elliptic curve cryptographic
More informationCHAPTER 5. Number Theory. 1. Integers and Division. Discussion
CHAPTER 5 Number Theory 1. Integers and Division 1.1. Divisibility. Definition 1.1.1. Given two integers a and b we say a divides b if there is an integer c such that b = ac. If a divides b, we write a
More informationPrime numbers and prime polynomials. Paul Pollack Dartmouth College
Prime numbers and prime polynomials Paul Pollack Dartmouth College May 1, 2008 Analogies everywhere! Analogies in elementary number theory (continued fractions, quadratic reciprocity, Fermat s last theorem)
More informationNetwork Protocol Design and Evaluation
Network Protocol Design and Evaluation 07 - Simulation, Part I Stefan Rührup Summer 2009 Overview In the last chapters: Formal Specification, Validation, Design Techniques Implementation Software Engineering,
More informationIntroduction to Queueing Theory and Stochastic Teletraffic Models
Introduction to Queueing Theory and Stochastic Teletraffic Models Moshe Zukerman EE Department, City University of Hong Kong Copyright M. Zukerman c 2000 2015 Preface The aim of this textbook is to provide
More informationGenerating Random Variables and Stochastic Processes
Monte Carlo Simulation: IEOR E4703 Fall 2004 c 2004 by Martin Haugh Generating Random Variables and Stochastic Processes 1 Generating U(0,1) Random Variables The ability to generate U(0, 1) random variables
More informationI. GROUPS: BASIC DEFINITIONS AND EXAMPLES
I GROUPS: BASIC DEFINITIONS AND EXAMPLES Definition 1: An operation on a set G is a function : G G G Definition 2: A group is a set G which is equipped with an operation and a special element e G, called
More informationUNIT I: RANDOM VARIABLES PART- A -TWO MARKS
UNIT I: RANDOM VARIABLES PART- A -TWO MARKS 1. Given the probability density function of a continuous random variable X as follows f(x) = 6x (1-x) 0
More informationRandom variables, probability distributions, binomial random variable
Week 4 lecture notes. WEEK 4 page 1 Random variables, probability distributions, binomial random variable Eample 1 : Consider the eperiment of flipping a fair coin three times. The number of tails that
More informationLecture 6: Discrete & Continuous Probability and Random Variables
Lecture 6: Discrete & Continuous Probability and Random Variables D. Alex Hughes Math Camp September 17, 2015 D. Alex Hughes (Math Camp) Lecture 6: Discrete & Continuous Probability and Random September
More informationGenerating Random Numbers Variance Reduction Quasi-Monte Carlo. Simulation Methods. Leonid Kogan. MIT, Sloan. 15.450, Fall 2010
Simulation Methods Leonid Kogan MIT, Sloan 15.450, Fall 2010 c Leonid Kogan ( MIT, Sloan ) Simulation Methods 15.450, Fall 2010 1 / 35 Outline 1 Generating Random Numbers 2 Variance Reduction 3 Quasi-Monte
More informationNetwork Security. Chapter 6 Random Number Generation
Network Security Chapter 6 Random Number Generation 1 Tasks of Key Management (1)! Generation:! It is crucial to security, that keys are generated with a truly random or at least a pseudo-random generation
More informationContinued Fractions and the Euclidean Algorithm
Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction
More informationCOMP 250 Fall 2012 lecture 2 binary representations Sept. 11, 2012
Binary numbers The reason humans represent numbers using decimal (the ten digits from 0,1,... 9) is that we have ten fingers. There is no other reason than that. There is nothing special otherwise about
More informationHomework until Test #2
MATH31: Number Theory Homework until Test # Philipp BRAUN Section 3.1 page 43, 1. It has been conjectured that there are infinitely many primes of the form n. Exhibit five such primes. Solution. Five such
More information1 Sufficient statistics
1 Sufficient statistics A statistic is a function T = rx 1, X 2,, X n of the random sample X 1, X 2,, X n. Examples are X n = 1 n s 2 = = X i, 1 n 1 the sample mean X i X n 2, the sample variance T 1 =
More informationInternet Dial-Up Traffic Modelling
NTS 5, Fifteenth Nordic Teletraffic Seminar Lund, Sweden, August 4, Internet Dial-Up Traffic Modelling Villy B. Iversen Arne J. Glenstrup Jens Rasmussen Abstract This paper deals with analysis and modelling
More informationSystems of Linear Equations
Systems of Linear Equations Beifang Chen Systems of linear equations Linear systems A linear equation in variables x, x,, x n is an equation of the form a x + a x + + a n x n = b, where a, a,, a n and
More informationTHE NUMBER OF REPRESENTATIONS OF n OF THE FORM n = x 2 2 y, x > 0, y 0
THE NUMBER OF REPRESENTATIONS OF n OF THE FORM n = x 2 2 y, x > 0, y 0 RICHARD J. MATHAR Abstract. We count solutions to the Ramanujan-Nagell equation 2 y +n = x 2 for fixed positive n. The computational
More informationStochastic Processes and Queueing Theory used in Cloud Computer Performance Simulations
56 Stochastic Processes and Queueing Theory used in Cloud Computer Performance Simulations Stochastic Processes and Queueing Theory used in Cloud Computer Performance Simulations Florin-Cătălin ENACHE
More informationE3: PROBABILITY AND STATISTICS lecture notes
E3: PROBABILITY AND STATISTICS lecture notes 2 Contents 1 PROBABILITY THEORY 7 1.1 Experiments and random events............................ 7 1.2 Certain event. Impossible event............................
More informationQuotient Rings and Field Extensions
Chapter 5 Quotient Rings and Field Extensions In this chapter we describe a method for producing field extension of a given field. If F is a field, then a field extension is a field K that contains F.
More informationDepartment of Mathematics, Indian Institute of Technology, Kharagpur Assignment 2-3, Probability and Statistics, March 2015. Due:-March 25, 2015.
Department of Mathematics, Indian Institute of Technology, Kharagpur Assignment -3, Probability and Statistics, March 05. Due:-March 5, 05.. Show that the function 0 for x < x+ F (x) = 4 for x < for x
More informationU.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009. Notes on Algebra
U.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009 Notes on Algebra These notes contain as little theory as possible, and most results are stated without proof. Any introductory
More information4 Sums of Random Variables
Sums of a Random Variables 47 4 Sums of Random Variables Many of the variables dealt with in physics can be expressed as a sum of other variables; often the components of the sum are statistically independent.
More informationFor a partition B 1,..., B n, where B i B j = for i. A = (A B 1 ) (A B 2 ),..., (A B n ) and thus. P (A) = P (A B i ) = P (A B i )P (B i )
Probability Review 15.075 Cynthia Rudin A probability space, defined by Kolmogorov (1903-1987) consists of: A set of outcomes S, e.g., for the roll of a die, S = {1, 2, 3, 4, 5, 6}, 1 1 2 1 6 for the roll
More information1.5 / 1 -- Communication Networks II (Görg) -- www.comnets.uni-bremen.de. 1.5 Transforms
.5 / -- Communication Networks II (Görg) -- www.comnets.uni-bremen.de.5 Transforms Using different summation and integral transformations pmf, pdf and cdf/ccdf can be transformed in such a way, that even
More informationRandom-Number Generation
Random-Number Generation Raj Jain Washington University Saint Louis, MO 63130 Jain@cse.wustl.edu Audio/Video recordings of this lecture are available at: http://www.cse.wustl.edu/~jain/cse574-08/ 26-1
More informationGenerating Random Variables and Stochastic Processes
Monte Carlo Simulation: IEOR E4703 c 2010 by Martin Haugh Generating Random Variables and Stochastic Processes In these lecture notes we describe the principal methods that are used to generate random
More informationNetwork Security. Chapter 6 Random Number Generation. Prof. Dr.-Ing. Georg Carle
Network Security Chapter 6 Random Number Generation Prof. Dr.-Ing. Georg Carle Chair for Computer Networks & Internet Wilhelm-Schickard-Institute for Computer Science University of Tübingen http://net.informatik.uni-tuebingen.de/
More information5: Magnitude 6: Convert to Polar 7: Convert to Rectangular
TI-NSPIRE CALCULATOR MENUS 1: Tools > 1: Define 2: Recall Definition --------------- 3: Delete Variable 4: Clear a-z 5: Clear History --------------- 6: Insert Comment 2: Number > 1: Convert to Decimal
More information1 Prior Probability and Posterior Probability
Math 541: Statistical Theory II Bayesian Approach to Parameter Estimation Lecturer: Songfeng Zheng 1 Prior Probability and Posterior Probability Consider now a problem of statistical inference in which
More information1 if 1 x 0 1 if 0 x 1
Chapter 3 Continuity In this chapter we begin by defining the fundamental notion of continuity for real valued functions of a single real variable. When trying to decide whether a given function is or
More informationWhat is Statistics? Lecture 1. Introduction and probability review. Idea of parametric inference
0. 1. Introduction and probability review 1.1. What is Statistics? What is Statistics? Lecture 1. Introduction and probability review There are many definitions: I will use A set of principle and procedures
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 5 9/17/2008 RANDOM VARIABLES
MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 5 9/17/2008 RANDOM VARIABLES Contents 1. Random variables and measurable functions 2. Cumulative distribution functions 3. Discrete
More informationLecture 3: Continuous distributions, expected value & mean, variance, the normal distribution
Lecture 3: Continuous distributions, expected value & mean, variance, the normal distribution 8 October 2007 In this lecture we ll learn the following: 1. how continuous probability distributions differ
More informationSMT 2014 Algebra Test Solutions February 15, 2014
1. Alice and Bob are painting a house. If Alice and Bob do not take any breaks, they will finish painting the house in 20 hours. If, however, Bob stops painting once the house is half-finished, then the
More informationChapter G08 Nonparametric Statistics
G08 Nonparametric Statistics Chapter G08 Nonparametric Statistics Contents 1 Scope of the Chapter 2 2 Background to the Problems 2 2.1 Parametric and Nonparametric Hypothesis Testing......................
More informationThis document is published at www.agner.org/random, Feb. 2008, as part of a software package.
Sampling methods by Agner Fog This document is published at www.agner.org/random, Feb. 008, as part of a software package. Introduction A C++ class library of non-uniform random number generators is available
More informationCopy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any.
Algebra 2 - Chapter Prerequisites Vocabulary Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any. P1 p. 1 1. counting(natural) numbers - {1,2,3,4,...}
More informationBinomial lattice model for stock prices
Copyright c 2007 by Karl Sigman Binomial lattice model for stock prices Here we model the price of a stock in discrete time by a Markov chain of the recursive form S n+ S n Y n+, n 0, where the {Y i }
More informationMaster s Theory Exam Spring 2006
Spring 2006 This exam contains 7 questions. You should attempt them all. Each question is divided into parts to help lead you through the material. You should attempt to complete as much of each problem
More informationBusiness Statistics. Successful completion of Introductory and/or Intermediate Algebra courses is recommended before taking Business Statistics.
Business Course Text Bowerman, Bruce L., Richard T. O'Connell, J. B. Orris, and Dawn C. Porter. Essentials of Business, 2nd edition, McGraw-Hill/Irwin, 2008, ISBN: 978-0-07-331988-9. Required Computing
More information1 Review of Newton Polynomials
cs: introduction to numerical analysis 0/0/0 Lecture 8: Polynomial Interpolation: Using Newton Polynomials and Error Analysis Instructor: Professor Amos Ron Scribes: Giordano Fusco, Mark Cowlishaw, Nathanael
More informationA Primer on Mathematical Statistics and Univariate Distributions; The Normal Distribution; The GLM with the Normal Distribution
A Primer on Mathematical Statistics and Univariate Distributions; The Normal Distribution; The GLM with the Normal Distribution PSYC 943 (930): Fundamentals of Multivariate Modeling Lecture 4: September
More informationFactoring Algorithms
Factoring Algorithms The p 1 Method and Quadratic Sieve November 17, 2008 () Factoring Algorithms November 17, 2008 1 / 12 Fermat s factoring method Fermat made the observation that if n has two factors
More informationOverview of Violations of the Basic Assumptions in the Classical Normal Linear Regression Model
Overview of Violations of the Basic Assumptions in the Classical Normal Linear Regression Model 1 September 004 A. Introduction and assumptions The classical normal linear regression model can be written
More informationQueuing Theory. Long Term Averages. Assumptions. Interesting Values. Queuing Model
Queuing Theory Queuing Theory Queuing theory is the mathematics of waiting lines. It is extremely useful in predicting and evaluating system performance. Queuing theory has been used for operations research.
More informationMATH BOOK OF PROBLEMS SERIES. New from Pearson Custom Publishing!
MATH BOOK OF PROBLEMS SERIES New from Pearson Custom Publishing! The Math Book of Problems Series is a database of math problems for the following courses: Pre-algebra Algebra Pre-calculus Calculus Statistics
More informationLecture Notes 1. Brief Review of Basic Probability
Probability Review Lecture Notes Brief Review of Basic Probability I assume you know basic probability. Chapters -3 are a review. I will assume you have read and understood Chapters -3. Here is a very
More informationminimal polyonomial Example
Minimal Polynomials Definition Let α be an element in GF(p e ). We call the monic polynomial of smallest degree which has coefficients in GF(p) and α as a root, the minimal polyonomial of α. Example: We
More informationCurriculum Map Statistics and Probability Honors (348) Saugus High School Saugus Public Schools 2009-2010
Curriculum Map Statistics and Probability Honors (348) Saugus High School Saugus Public Schools 2009-2010 Week 1 Week 2 14.0 Students organize and describe distributions of data by using a number of different
More informationChapter 3. if 2 a i then location: = i. Page 40
Chapter 3 1. Describe an algorithm that takes a list of n integers a 1,a 2,,a n and finds the number of integers each greater than five in the list. Ans: procedure greaterthanfive(a 1,,a n : integers)
More information5. Continuous Random Variables
5. Continuous Random Variables Continuous random variables can take any value in an interval. They are used to model physical characteristics such as time, length, position, etc. Examples (i) Let X be
More informationDefinition: Suppose that two random variables, either continuous or discrete, X and Y have joint density
HW MATH 461/561 Lecture Notes 15 1 Definition: Suppose that two random variables, either continuous or discrete, X and Y have joint density and marginal densities f(x, y), (x, y) Λ X,Y f X (x), x Λ X,
More informationLecture 2: Universality
CS 710: Complexity Theory 1/21/2010 Lecture 2: Universality Instructor: Dieter van Melkebeek Scribe: Tyson Williams In this lecture, we introduce the notion of a universal machine, develop efficient universal
More informationLoad Balancing and Switch Scheduling
EE384Y Project Final Report Load Balancing and Switch Scheduling Xiangheng Liu Department of Electrical Engineering Stanford University, Stanford CA 94305 Email: liuxh@systems.stanford.edu Abstract Load
More information. (3.3) n Note that supremum (3.2) must occur at one of the observed values x i or to the left of x i.
Chapter 3 Kolmogorov-Smirnov Tests There are many situations where experimenters need to know what is the distribution of the population of their interest. For example, if they want to use a parametric
More information7 Gaussian Elimination and LU Factorization
7 Gaussian Elimination and LU Factorization In this final section on matrix factorization methods for solving Ax = b we want to take a closer look at Gaussian elimination (probably the best known method
More informationEvaluating the Lead Time Demand Distribution for (r, Q) Policies Under Intermittent Demand
Proceedings of the 2009 Industrial Engineering Research Conference Evaluating the Lead Time Demand Distribution for (r, Q) Policies Under Intermittent Demand Yasin Unlu, Manuel D. Rossetti Department of
More informationHow To Understand And Solve A Linear Programming Problem
At the end of the lesson, you should be able to: Chapter 2: Systems of Linear Equations and Matrices: 2.1: Solutions of Linear Systems by the Echelon Method Define linear systems, unique solution, inconsistent,
More informationNEW YORK STATE TEACHER CERTIFICATION EXAMINATIONS
NEW YORK STATE TEACHER CERTIFICATION EXAMINATIONS TEST DESIGN AND FRAMEWORK September 2014 Authorized for Distribution by the New York State Education Department This test design and framework document
More informationThis unit will lay the groundwork for later units where the students will extend this knowledge to quadratic and exponential functions.
Algebra I Overview View unit yearlong overview here Many of the concepts presented in Algebra I are progressions of concepts that were introduced in grades 6 through 8. The content presented in this course
More informationCovariance and Correlation
Covariance and Correlation ( c Robert J. Serfling Not for reproduction or distribution) We have seen how to summarize a data-based relative frequency distribution by measures of location and spread, such
More informationNormal distribution. ) 2 /2σ. 2π σ
Normal distribution The normal distribution is the most widely known and used of all distributions. Because the normal distribution approximates many natural phenomena so well, it has developed into a
More informationFactoring & Primality
Factoring & Primality Lecturer: Dimitris Papadopoulos In this lecture we will discuss the problem of integer factorization and primality testing, two problems that have been the focus of a great amount
More informationCourse Text. Required Computing Software. Course Description. Course Objectives. StraighterLine. Business Statistics
Course Text Business Statistics Lind, Douglas A., Marchal, William A. and Samuel A. Wathen. Basic Statistics for Business and Economics, 7th edition, McGraw-Hill/Irwin, 2010, ISBN: 9780077384470 [This
More informationM2S1 Lecture Notes. G. A. Young http://www2.imperial.ac.uk/ ayoung
M2S1 Lecture Notes G. A. Young http://www2.imperial.ac.uk/ ayoung September 2011 ii Contents 1 DEFINITIONS, TERMINOLOGY, NOTATION 1 1.1 EVENTS AND THE SAMPLE SPACE......................... 1 1.1.1 OPERATIONS
More information1 The Brownian bridge construction
The Brownian bridge construction The Brownian bridge construction is a way to build a Brownian motion path by successively adding finer scale detail. This construction leads to a relatively easy proof
More informationNAG C Library Chapter Introduction. g08 Nonparametric Statistics
g08 Nonparametric Statistics Introduction g08 NAG C Library Chapter Introduction g08 Nonparametric Statistics Contents 1 Scope of the Chapter... 2 2 Background to the Problems... 2 2.1 Parametric and Nonparametric
More informationReview of Random Variables
Chapter 1 Review of Random Variables Updated: January 16, 2015 This chapter reviews basic probability concepts that are necessary for the modeling and statistical analysis of financial data. 1.1 Random
More informationMonte Carlo Methods in Finance
Author: Yiyang Yang Advisor: Pr. Xiaolin Li, Pr. Zari Rachev Department of Applied Mathematics and Statistics State University of New York at Stony Brook October 2, 2012 Outline Introduction 1 Introduction
More informationSections 2.11 and 5.8
Sections 211 and 58 Timothy Hanson Department of Statistics, University of South Carolina Stat 704: Data Analysis I 1/25 Gesell data Let X be the age in in months a child speaks his/her first word and
More information1) Write the following as an algebraic expression using x as the variable: Triple a number subtracted from the number
1) Write the following as an algebraic expression using x as the variable: Triple a number subtracted from the number A. 3(x - x) B. x 3 x C. 3x - x D. x - 3x 2) Write the following as an algebraic expression
More informationCurrent Standard: Mathematical Concepts and Applications Shape, Space, and Measurement- Primary
Shape, Space, and Measurement- Primary A student shall apply concepts of shape, space, and measurement to solve problems involving two- and three-dimensional shapes by demonstrating an understanding of:
More informationSUM OF TWO SQUARES JAHNAVI BHASKAR
SUM OF TWO SQUARES JAHNAVI BHASKAR Abstract. I will investigate which numbers can be written as the sum of two squares and in how many ways, providing enough basic number theory so even the unacquainted
More information