Lecture 6: Discrete & Continuous Probability and Random Variables
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1 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 Variables 17, / 33
2 1 Finishing Basics of Expectation and Variance Variance of Discrete Random Variables 2 Continuous Probability Introduction Continuous Probability Densities Cumulative Density Function Expectation of Continuous Random Variables Variance of Continuous Random Variables 3 Properties of Expecctation and Variance Properties of Expectation Properties of Variance D. Alex Hughes (Math Camp) Lecture 6: Discrete & Continuous Probability and Random September Variables 17, / 33
3 Variance of Discrete Random Variables Definition The variance of a random variable is the expected value of its squared deviations from µ. If X is discrete with pdf p X (k), VAR[X ] = σ 2 = E[(X µ) 2 ] = k (k µ) 2 p X (k). Note that µ = E[X ]. Also note that the standard deviation σ is the square root of the variance. VAR[X ] = E[(X E[X ]) 2 ] Just as the expected value has a clear analogy to the center of balance of a distribution, the variance is analogous to the moment of inertia. Imagine placing two distributions on a turntable and pushing on each with identical force 6 from the center. A distribution with low variance would spin faster, and a distribution with high variance would spin slower. D. Alex Hughes (Math Camp) Lecture 6: Discrete & Continuous Probability and Random September Variables 17, / 33
4 Discrete RV Variance Example What is the variance of a six sided die roll? D. Alex Hughes (Math Camp) Lecture 6: Discrete & Continuous Probability and Random September Variables 17, / 33
5 Discrete RV Variance Example What is the variance of a six sided die roll? Answer First calculate the expected value of a single die roll. 6 x i p(x i ) = 1(1/6) + 2(1/6) + 3(1/6) 1 + 4(1/6) + 5(1/6) + 6(1/6) E[X ] = µ = 3.5 D. Alex Hughes (Math Camp) Lecture 6: Discrete & Continuous Probability and Random September Variables 17, / 33
6 Discrete RV Variance Answer Then the Variance is: 6 (x i µ) 2 p(x i ) = (1 3.5) 2 (1/6) + (2 3.5) 2 (1/6) + (3 3.5) 2 (1/6) 1 + (4 3.5) 2 (1/6) + (5 3.5) 2 (1/6) + (6 3.5) 2 (1/6) = 17.5(1/6) = 2.92 D. Alex Hughes (Math Camp) Lecture 6: Discrete & Continuous Probability and Random September Variables 17, / 33
7 Example of Discrete RV Variance Example An urn contains five chips, two red and three white. Two chips are drawn at random, with replacement. Let X denote the number of red chips drawn. What is VAR[X]? D. Alex Hughes (Math Camp) Lecture 6: Discrete & Continuous Probability and Random September Variables 17, / 33
8 Example of Discrete RV Variance Example An urn contains five chips, two red and three white. Two chips are drawn at random, with replacement. Let X denote the number of red chips drawn. What is VAR[X]? Answer X is binomial, n = 2 and p = 2 5. E[X ] = np = (2)(2/5) = 4/5 VAR[X ] = 2 ( ) ( ) (x E[X ]) 2 f (x) = (i.8) i ( ) 3 2 i i 5 5 i=0 ( ) ( ) ( ) = (0.8) 2 2 (2/5) 0 (3/5) 2 + (1.8) 2 2 (2/5) 1 (3/5) 1 + (2.8) 2 2 (2/5) 2 (3/5) = (16/25)(1)(9/25) + (1/25)(2)(6/25) + (36/25)(1)(4/25) = D. Alex Hughes (Math Camp) Lecture 6: Discrete & Continuous Probability and Random September Variables 17, / 33
9 1 Finishing Basics of Expectation and Variance Variance of Discrete Random Variables 2 Continuous Probability Introduction Continuous Probability Densities Cumulative Density Function Expectation of Continuous Random Variables Variance of Continuous Random Variables 3 Properties of Expecctation and Variance Properties of Expectation Properties of Variance D. Alex Hughes (Math Camp) Lecture 6: Discrete & Continuous Probability and Random September Variables 17, / 33
10 Example Example Some process has equal chance of generating a number between [0,1]. What is the probability the process generate a number less that 0.5? D. Alex Hughes (Math Camp) Lecture 6: Discrete & Continuous Probability and Random September Variables 17, / 33
11 Example Example Some process has equal chance of generating a number between [0,1]. What is the probability the process generate a number less that 0.5? Draw a number line. D. Alex Hughes (Math Camp) Lecture 6: Discrete & Continuous Probability and Random September Variables 17, / 33
12 Example Example Some process has equal chance of generating a number between [0,1]. What is the probability the process generate a number less that 0.5? Draw a number line. Probability Outcome Value D. Alex Hughes (Math Camp) Lecture 6: Discrete & Continuous Probability and Random September Variables 17, / 33
13 Example Example Some process has equal chance of generating a number between [0,1]. What is the probability the process generate a number less that 0.5? Draw a number line. Probability Outcome Value D. Alex Hughes (Math Camp) Lecture 6: Discrete & Continuous Probability and Random September Variables 17, / 33
14 Example Example Some process has equal chance of generating a number between [0,1]. What is the probability the process generate a number less that 0.5? Draw a number line. Probability Outcome Value D. Alex Hughes (Math Camp) Lecture 6: Discrete & Continuous Probability and Random September Variables 17, / 33
15 Continuous Probability Definition A probability density function P on a set of real numbers S is called Continuous is there exists a function f(t) such that for any closed interval [a,b] S, P([a, b]) = b a f (t)dt D. Alex Hughes (Math Camp) Lecture 6: Discrete & Continuous Probability and Random September Variables 17, / 33
16 Continuous Probability Definition A probability density function P on a set of real numbers S is called Continuous is there exists a function f(t) such that for any closed interval [a,b] S, Comment f(t) must satisfy two properties: P([a, b]) = b a f (t)dt 1 f (t) 0 for all t 2 f (t)dt = 1 Frequently, we refer to the probability distribution function as the pdf. D. Alex Hughes (Math Camp) Lecture 6: Discrete & Continuous Probability and Random September Variables 17, / 33
17 Continuous Probability Example Example What continuous probability function describes the uniform chance of drawing a number on the range [0,10]? What is the probability that a randomly selected number falls between 4 and 7? What is the probability that a randomly selected number falls between [2,4] or [6,9]? What continuous probability function describes the uniform chance of drawing a number on the range [a,b]? D. Alex Hughes (Math Camp) Lecture 6: Discrete & Continuous Probability and Random September Variables 17, / 33
18 Continuous Probability Example 2 Example Is f (x) = 3x 2, 0 x 1 a valid probability function? D. Alex Hughes (Math Camp) Lecture 6: Discrete & Continuous Probability and Random September Variables 17, / 33
19 Continuous Probability Example 2 Example Is f (x) = 3x 2, 0 x 1 a valid probability function? What is the probability that 0 x 1/3? What is the probability that 2/3 x 1? D. Alex Hughes (Math Camp) Lecture 6: Discrete & Continuous Probability and Random September Variables 17, / 33
20 Continuous Probability Example 2 Example Is f (x) = 3x 2, 0 x 1 a valid probability function? What is the probability that 0 x 1/3? What is the probability that 2/3 x 1? Answer Yes. Recall the two properties f(x) just satisfy: (A) f(x) is always non-negative; (B) The integral of f(x) across the function s domain = 1. 1/3 0 = x 3 1/3 0 = 1/27 1 2/3 = x 3 1 2/3 = 1 8/27 = 19/27. D. Alex Hughes (Math Camp) Lecture 6: Discrete & Continuous Probability and Random September Variables 17, / 33
21 Example 3 Example Suppose we would like a continuous function to describe a variable Y in such a way that it more likely to produce y s near the middle of the range more frequently than near the extremes of the range. What might this look like? D. Alex Hughes (Math Camp) Lecture 6: Discrete & Continuous Probability and Random September Variables 17, / 33
22 Example 3 Example Suppose we would like a continuous function to describe a variable Y in such a way that it more likely to produce y s near the middle of the range more frequently than near the extremes of the range. What might this look like? f(y) I ve drawn one where f (y) = 6y(1 y) y D. Alex Hughes (Math Camp) Lecture 6: Discrete & Continuous Probability and Random September Variables 17, / 33
23 Example 3 Example Does this meet the requirements of a probability density function? D. Alex Hughes (Math Camp) Lecture 6: Discrete & Continuous Probability and Random September Variables 17, / 33
24 Example 3 Example Does this meet the requirements of a probability density function? Answer Yes! 1 f Y 0 for all y; 2 P(Ω) = 1. Then, with this function, answer the following: What is the probability that a number is less than 0.5? Greater than 0.5? Less than 0.25? Greater than 0.25? Exactly 0.25? D. Alex Hughes (Math Camp) Lecture 6: Discrete & Continuous Probability and Random September Variables 17, / 33
25 Cumulative Density Function Definition The cumulative density function (called cdf ) of a continuous, real-valued random variable Y is an indefinite integral of its pdf. F Y (x) = x f y (x)dx = P(X x) Note the difference in notation between a cumulative distribution function (cdf) and a probability distribution function (pdf). Theorem F (y) = y d F (y) = f (y) dy f (t)dt; conversely, D. Alex Hughes (Math Camp) Lecture 6: Discrete & Continuous Probability and Random September Variables 17, / 33
26 Cumulative Density Examples Example A uniform distribution covers the range [0,10]. What is the probability that a random number drawn is less than 4? D. Alex Hughes (Math Camp) Lecture 6: Discrete & Continuous Probability and Random September Variables 17, / 33
27 Cumulative Density Examples Example A uniform distribution covers the range [0,10]. What is the probability that a random number drawn is less than 4? Illustration of Cumulative Density Function y Area = 0.4 Point = f (x)dx =.1x 4 0 =.1(4).1(0) f (x) = F X (x) F X (x) = 0.1x F X (4) = 0.1(4) = x D. Alex Hughes (Math Camp) Lecture 6: Discrete & Continuous Probability and Random September Variables 17, / 33
28 CDF Properties Theorem Let Y be a continuous random variable with cdf F Y (y), Then, 1 P(Y > s) = 1 F Y (s) 2 P(r < Y s) = F Y (s) F Y (r) 3 lim x + F Y (y) = 1 4 lim x F Y (y) = 0 Additionally: The CDF is a monotonically nondecreasing continuous function. Will be useful for easily/quickly calculating the probability a variable takes a value on some interval. D. Alex Hughes (Math Camp) Lecture 6: Discrete & Continuous Probability and Random September Variables 17, / 33
29 An Example In Encinitas the distribution of waves a broski, Tad, takes every hour, Y, is described by the pdf: f Y (y) = y e y, y 0. What is the median of Encinitas wave-income distribution? D. Alex Hughes (Math Camp) Lecture 6: Discrete & Continuous Probability and Random September Variables 17, / 33
30 An Example In Encinitas the distribution of waves a broski, Tad, takes every hour, Y, is described by the pdf: f Y (y) = y e y, y 0. What is the median of Encinitas wave-income distribution? In looking for the median, we are looking for the number where the probability below is equal to the probability above. D. Alex Hughes (Math Camp) Lecture 6: Discrete & Continuous Probability and Random September Variables 17, / 33
31 An Example In Encinitas the distribution of waves a broski, Tad, takes every hour, Y, is described by the pdf: f Y (y) = y e y, y 0. What is the median of Encinitas wave-income distribution? In looking for the median, we are looking for the number where the probability below is equal to the probability above. F Y (y) = ye y 0 = e y 0.5 = e y y 0.5 = 0 ( e y ) 0.5 = e y ln(0.5) = y 0.69 D. Alex Hughes (Math Camp) Lecture 6: Discrete & Continuous Probability and Random September Variables 17, / 33
32 Expectation of Continuous Random Variables Definition If Y is a continuous random variable with pdf f Y (y), E[Y ] = y f Y (y)dy In the same was as the discrete case, one may think of this as a center of gravity of the distribution. D. Alex Hughes (Math Camp) Lecture 6: Discrete & Continuous Probability and Random September Variables 17, / 33
33 Example A continuous uniform density function, f(x) = 1 is defined on the range [0,1]. What is the expected value of this function? Answer E[X ] = = = 1 2 x x f (x)dx 1xdx = 1 (1 0) 2 = 0.5 D. Alex Hughes (Math Camp) Lecture 6: Discrete & Continuous Probability and Random September Variables 17, / 33
34 Variance of Continuous Random Variables Definition Let X be a real-valued random variable with density function f X (x). Then, the variance, σ 2, is defined by σ 2 = E[(X µ) 2 ] = (x µ) 2 f (x)dx D. Alex Hughes (Math Camp) Lecture 6: Discrete & Continuous Probability and Random September Variables 17, / 33
35 Expectation Identities Theorem For any constant c, E[c] = c D. Alex Hughes (Math Camp) Lecture 6: Discrete & Continuous Probability and Random September Variables 17, / 33
36 Expectation Identities Theorem For any constant c, E[c] = c Theorem For any random variable W, E[W + b] = E[W ] + b D. Alex Hughes (Math Camp) Lecture 6: Discrete & Continuous Probability and Random September Variables 17, / 33
37 Expectation Identities Theorem For any constant c, E[c] = c Theorem For any random variable W, E[W + b] = E[W ] + b Theorem For any constant c and random variable X, E[cX ] = ce[x ] D. Alex Hughes (Math Camp) Lecture 6: Discrete & Continuous Probability and Random September Variables 17, / 33
38 Expectation Identities Theorem For any constant c, E[c] = c Theorem For any random variable W, E[W + b] = E[W ] + b Theorem For any constant c and random variable X, E[cX ] = ce[x ] Theorem For any X and Y that are random variables that have finite expectations E[X + Y ] = E[X ] + E[Y ] D. Alex Hughes (Math Camp) Lecture 6: Discrete & Continuous Probability and Random September Variables 17, / 33
39 Expected Value of a Function of a Random Variable Theorem Suppose X is a random variable with pdf p X (k). Let g(x ) be any function of X. Then the expected value of the random variable g(x) is given by Example E[g(X )] = g(x) f X (x)dx Suppose X = f X (x) is the uniform density function defined on the range [0,1]. Then f X (x) = 1. Suppose g(x) = 1/5x. E[g(X )] = x f X (x)dx = 1 5 [1 2 x 2 ] 1 0 = 1 10 D. Alex Hughes (Math Camp) Lecture 6: Discrete & Continuous Probability and Random September Variables 17, / 33
40 Example The Zetas want to burn their gang-signs into a wall. Suppose the amount of fuel in the torch, Y, is a random variable with pdf f Y (y) = 3y 2, 0 < y < 1. In the past they have been able to burn a circle whose radius is 20 times the size of y. How much area can they expect to burn? D. Alex Hughes (Math Camp) Lecture 6: Discrete & Continuous Probability and Random September Variables 17, / 33
41 Example The Zetas want to burn their gang-signs into a wall. Suppose the amount of fuel in the torch, Y, is a random variable with pdf f Y (y) = 3y 2, 0 < y < 1. In the past they have been able to burn a circle whose radius is 20 times the size of y. How much area can they expect to burn? Answer By problem setup, g(y ) = 20πY 2 E[g(Y )] = 1 0 = 60πy 5 5 = 12πft 2 20πy 2 3y 2 dy = 60π y 4 0 D. Alex Hughes (Math Camp) Lecture 6: Discrete & Continuous Probability and Random September Variables 17, / 33
42 Variance Recall the definition of Variance: Definition The variance of a random variable is the expected value of its squared deviations from µ. If X is discrete with pdf p X (k), VAR[X ] = σ 2 = E[(X µ) 2 ] = k (k µ) 2 p X (k). Note that µ = E[X ]. Also note that the standard deviation σ is the square root of the variance. VAR[X ] = E[(X E[X ]) 2 ] D. Alex Hughes (Math Camp) Lecture 6: Discrete & Continuous Probability and Random September Variables 17, / 33
43 A Variance Identity Theorem Let X be any random variable (discrete or continuous), having mean µ and for which E[X 2 ] is finite. Then, Proof. VAR[X ] = E[(X E[X ]) 2 ] = E[X 2 ] E[X ] 2 VAR[X ] = E[(X E[X ]) 2 ] = E[X 2 2XE[X ] + E[X ] 2 ] = E[X 2 ] E[2XE[X ]] + E[E[X ] 2 ] = E[X 2 ] E[2X ]E[X ] + E[X ] 2 = E[X 2 ] 2E[X ]E[X ] + E[X ] 2 = E[X 2 ] E[X ] 2 D. Alex Hughes (Math Camp) Lecture 6: Discrete & Continuous Probability and Random September Variables 17, / 33
44 An Example Example Given the following pdf, what is the variance? f (x) = 3(1 x) 2, 0 < x < 1 D. Alex Hughes (Math Camp) Lecture 6: Discrete & Continuous Probability and Random September Variables 17, / 33
45 An Example Example Given the following pdf, what is the variance? Answer E[X ] = E[X 2 ] = f (x) = 3(1 x) 2, 0 < x < 1 x 3(1 x) 2 dx = 3 x 2 3(1 x) 2 dx = 3 VAR[X ] = E[X 2 ] E[X ] 2 = x 2y 2 + y 3 = 3 y 2 2y 3 + y 4 dx = 1 10 D. Alex Hughes (Math Camp) Lecture 6: Discrete & Continuous Probability and Random September Variables 17, / 33
46 Variance Identities Theorem For and random variable X and constant c VAR[cX ] = c 2 VAR[X ] VAR[X + c] = VAR[X ] Proof. For ease, let E[X ] = µ. Then E[cX ] = cµ, and V [cx ] = E[(cX cµ) 2 ] = E[c 2 (X µ) 2 ] = c 2 E[(X µ) 2 ] = c 2 VAR[X ] D. Alex Hughes (Math Camp) Lecture 6: Discrete & Continuous Probability and Random September Variables 17, / 33
47 Variance Identities Theorem Let X and Y be two independent random variables. Then, VAR[X + Y ] = VAR[X ] + VAR[Y ] Proof. Let E[X] = a and E[Y] = b. VAR[X + Y ] = E[(X + Y ) 2 ] (a + b) 2 = E[X 2 ] + 2E[XY ] + E[Y 2 ] a 2 2ab b 2 = E[X 2 ] a 2 + E[Y 2 ] b 2 = VAR[X ] + VAR[Y ] D. Alex Hughes (Math Camp) Lecture 6: Discrete & Continuous Probability and Random September Variables 17, / 33
48 Introduction What do we require of a probability distribution/density model? It must be non-negative for all outcomes in its sample space It must sum or integrate to one across the sample space. By this definition, f (y) = y y 3 2, 0 y 1 is a valid pdf. f (y) 0, y0 1, and; 1 y y 3 2 dy = 1 But, how useful is this model? D. Alex Hughes (Math Camp) Lecture 6: Discrete & Continuous Probability and Random September Variables 17, / 33
49 A Model s Merit A pdf has utility as a probability model if it actually models the behavior of the real (or political) world. A surprisingly small number of distributions describe these real world outcomes. Many measurements/outcomes are the result of the same set of assumptions about the data generating process Number of fraud incidents Number of latrines built Feeding patterns of zebra muscles D. Alex Hughes (Math Camp) Lecture 6: Discrete & Continuous Probability and Random September Variables 17, / 33
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