Math/Stat 370: Engineering Statistics, Washington State University
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1 Math/Stat 370: Engineering Statistics, Washington State University Haijun Li Department of Mathematics Washington State University Week 2 Haijun Li Math/Stat 370: Engineering Statistics, Washington State University Week 2 1 / 23
2 Outline 1 Section 3-2: Random Events Notations Random Variables 2 Section 3-3: Probability 3 Section 3-4: Continuous Random Variables Haijun Li Math/Stat 370: Engineering Statistics, Washington State University Week 2 2 / 23
3 Notations Sample space Ω := {all possible outcomes ω of the underlying experiment}. Example: Toss a coin three times. What is the sample space? Let H = head, T = tail. Then Ω = {HHH, HHT, HTH, THH, TTH, THT, HTT, TTT }. Haijun Li Math/Stat 370: Engineering Statistics, Washington State University Week 2 3 / 23
4 Notations Sample space Ω := {all possible outcomes ω of the underlying experiment}. Example: Toss a coin three times. What is the sample space? Let H = head, T = tail. Then Ω = {HHH, HHT, HTH, THH, TTH, THT, HTT, TTT }. Example: Toss a coin until the first head appears. Ω = {H, TH, TTH, TTTH, TTTTH,..., } T.{{.. T} H,... }. k 0 Haijun Li Math/Stat 370: Engineering Statistics, Washington State University Week 2 3 / 23
5 Notations Sample space Ω := {all possible outcomes ω of the underlying experiment}. Example: Toss a coin three times. What is the sample space? Let H = head, T = tail. Then Ω = {HHH, HHT, HTH, THH, TTH, THT, HTT, TTT }. Example: Toss a coin until the first head appears. Ω = {H, TH, TTH, TTTH, TTTTH,..., } T.{{.. T} H,... }. k 0 Random event E: Any subset of a sample space. Impossible event: Empty set. Sure event: Sample space Ω itself. Haijun Li Math/Stat 370: Engineering Statistics, Washington State University Week 2 3 / 23
6 Composite Events Example: Toss a coin three times. Describe the following events. 1 E 1 : three heads occur. E 1 = {HHH}. Haijun Li Math/Stat 370: Engineering Statistics, Washington State University Week 2 4 / 23
7 Composite Events Example: Toss a coin three times. Describe the following events. 1 E 1 : three heads occur. E 1 = {HHH}. 2 E 2 : exactly two heads occur. E 2 = {THH, HTH, HHT }. Haijun Li Math/Stat 370: Engineering Statistics, Washington State University Week 2 4 / 23
8 Composite Events Example: Toss a coin three times. Describe the following events. 1 E 1 : three heads occur. E 1 = {HHH}. 2 E 2 : exactly two heads occur. E 2 = {THH, HTH, HHT }. 3 E 3 : at least one tail occurs. E 3 = {TTT, TTH, THT, HTT, THH, HHT, HTH}. Haijun Li Math/Stat 370: Engineering Statistics, Washington State University Week 2 4 / 23
9 Composite Events Example: Toss a coin three times. Describe the following events. 1 E 1 : three heads occur. E 1 = {HHH}. 2 E 2 : exactly two heads occur. E 2 = {THH, HTH, HHT }. 3 E 3 : at least one tail occurs. E 3 = {TTT, TTH, THT, HTT, THH, HHT, HTH}. Composite Events via Set Operations Union E 1 E 2 : E 1 or E 2 occurs. Intersection E 1 E 2 : Both E 1 and E 2 occur. Complement E : E does not occur. Mutually exclusive: E 1 E 2 =. Haijun Li Math/Stat 370: Engineering Statistics, Washington State University Week 2 4 / 23
10 Random Variables Random variable: A function defined on the sample space. Example: Toss a coin three times. Let N = number of heads in three tosses. N(TTH) = 1, N(HHH) = 3, N(HTH) = 2, N(TTT ) = 0. Example: Sample a product from an assembly line. Let T = lifelength of the item. Haijun Li Math/Stat 370: Engineering Statistics, Washington State University Week 2 5 / 23
11 Random Variables Random variable: A function defined on the sample space. Example: Toss a coin three times. Let N = number of heads in three tosses. N(TTH) = 1, N(HHH) = 3, N(HTH) = 2, N(TTT ) = 0. Example: Sample a product from an assembly line. Let T = lifelength of the item. Discrete random variable: Its values are limited to discrete points (i.e., finite or countably infinite) on the real line. Continuous random variable: It takes on continuous measurements. Haijun Li Math/Stat 370: Engineering Statistics, Washington State University Week 2 5 / 23
12 Probability Probability P(E): Likelihood of the random event E (relative frequency, or degree of belief). Haijun Li Math/Stat 370: Engineering Statistics, Washington State University Week 2 6 / 23
13 Probability Probability P(E): Likelihood of the random event E (relative frequency, or degree of belief). Basic Properties of Probability 1 0 P(E) 1. 2 P(Ω) = 1. mutually-exclusive-1.gif (GIF Image, pixels) 3 If E 1, E 2,..., E k are mutually exclusive, P(E 1 E 2 E k ) = P(E 1 ) + + P(E k ) = k i=1 P(E i). Haijun Li Math/Stat 370: Engineering Statistics, Washington State University Week 2 6 / 23
14 Interpreting Probability Probability P(E): Likelihood of the random event E (relative frequency, or degree of belief,...). Figure: p n i=1 x i/n, x i = 1 (head) x i = 0 (tail), n = number of trials Haijun Li Math/Stat 370: Engineering Statistics, Washington State University Week 2 7 / 23
15 More Properties P(A ) = 1 P(A). P( ) = 0. diagram_example_0.png For any events (PNG Image, B and 484 C, 243 pixels) P(B C) = P(B) + P(C) P(B C). Haijun Li Math/Stat 370: Engineering Statistics, Washington State University Week 2 8 / 23
16 Probability Associated to Random Variables Let A be a subset of real numbers, and symbolify belong to. Let X be a random variable. P(X A) = probability that X is in A. Example: P(X = 1), P(X 5), P(X > 2). Haijun Li Math/Stat 370: Engineering Statistics, Washington State University Week 2 9 / 23
17 Probability Associated to Random Variables Let A be a subset of real numbers, and symbolify belong to. Let X be a random variable. P(X A) = probability that X is in A. Example: P(X = 1), P(X 5), P(X > 2). Example: Let N be the number of heads in three tosses of a fair coin. Calculate P(N = 1) and P(N 2). Haijun Li Math/Stat 370: Engineering Statistics, Washington State University Week 2 9 / 23
18 Probability Associated to Random Variables Let A be a subset of real numbers, and symbolify belong to. Let X be a random variable. P(X A) = probability that X is in A. Example: P(X = 1), P(X 5), P(X > 2). Example: Let N be the number of heads in three tosses of a fair coin. Calculate P(N = 1) and P(N 2). P(N = 1) = 3/8. Haijun Li Math/Stat 370: Engineering Statistics, Washington State University Week 2 9 / 23
19 Probability Associated to Random Variables Let A be a subset of real numbers, and symbolify belong to. Let X be a random variable. P(X A) = probability that X is in A. Example: P(X = 1), P(X 5), P(X > 2). Example: Let N be the number of heads in three tosses of a fair coin. Calculate P(N = 1) and P(N 2). P(N = 1) = 3/8. P(N 2) = 1 P(N = 3) = 1 1/8 = 7/8. Haijun Li Math/Stat 370: Engineering Statistics, Washington State University Week 2 9 / 23
20 Probability Associated to Random Variables Let A be a subset of real numbers, and symbolify belong to. Let X be a random variable. P(X A) = probability that X is in A. Example: P(X = 1), P(X 5), P(X > 2). Example: Let N be the number of heads in three tosses of a fair coin. Calculate P(N = 1) and P(N 2). P(N = 1) = 3/8. P(N 2) = 1 P(N = 3) = 1 1/8 = 7/8. Example: If P(X A) = 0.4 and P(X B) = 0.5 and P(X A B) = 0.2. Find Haijun Li Math/Stat 370: Engineering Statistics, Washington State University Week 2 9 / 23
21 Probability Associated to Random Variables Let A be a subset of real numbers, and symbolify belong to. Let X be a random variable. P(X A) = probability that X is in A. Example: P(X = 1), P(X 5), P(X > 2). Example: Let N be the number of heads in three tosses of a fair coin. Calculate P(N = 1) and P(N 2). P(N = 1) = 3/8. P(N 2) = 1 P(N = 3) = 1 1/8 = 7/8. Example: If P(X A) = 0.4 and P(X B) = 0.5 and P(X A B) = 0.2. Find 1 P(X A ) = 1 P(X A) = 0.6. Haijun Li Math/Stat 370: Engineering Statistics, Washington State University Week 2 9 / 23
22 Probability Associated to Random Variables Let A be a subset of real numbers, and symbolify belong to. Let X be a random variable. P(X A) = probability that X is in A. Example: P(X = 1), P(X 5), P(X > 2). Example: Let N be the number of heads in three tosses of a fair coin. Calculate P(N = 1) and P(N 2). P(N = 1) = 3/8. P(N 2) = 1 P(N = 3) = 1 1/8 = 7/8. Example: If P(X A) = 0.4 and P(X B) = 0.5 and P(X A B) = 0.2. Find 1 P(X A ) = 1 P(X A) = P(X A B) = = 0.7. Haijun Li Math/Stat 370: Engineering Statistics, Washington State University Week 2 9 / 23
23 Example Let X denote the life of a semiconductor laser (in hours) with P(X 5000) = 0.05, P(X > 7000) = Find the following. 1 The probability that the life is greater than 5000 hours; Haijun Li Math/Stat 370: Engineering Statistics, Washington State University Week 2 10 / 23
24 Example Let X denote the life of a semiconductor laser (in hours) with Find the following. P(X 5000) = 0.05, P(X > 7000) = The probability that the life is greater than 5000 hours; P(X > 5000) = = Haijun Li Math/Stat 370: Engineering Statistics, Washington State University Week 2 10 / 23
25 Example Let X denote the life of a semiconductor laser (in hours) with Find the following. P(X 5000) = 0.05, P(X > 7000) = The probability that the life is greater than 5000 hours; P(X > 5000) = = P(5000 < X 7000) = = 0.5. Haijun Li Math/Stat 370: Engineering Statistics, Washington State University Week 2 10 / 23
26 Distribution of Random Variables Random Variables (RV): Denoted by capital letters, X, Y, N,... Haijun Li Math/Stat 370: Engineering Statistics, Washington State University Week 2 11 / 23
27 Distribution of Random Variables Random Variables (RV): Denoted by capital letters, X, Y, N,... Values of Random Variables: Denoted by small letters, x, y, n,... Haijun Li Math/Stat 370: Engineering Statistics, Washington State University Week 2 11 / 23
28 Distribution of Random Variables Random Variables (RV): Denoted by capital letters, X, Y, N,... Values of Random Variables: Denoted by small letters, x, y, n,... Distribution of RV X: Likelihoods or relative frequencies of various values of X. Haijun Li Math/Stat 370: Engineering Statistics, Washington State University Week 2 11 / 23
29 S&P 500 Monthly Returns (10/1965-7/2005) S&P 2.gif (GIF 500 Image, 572 Monthly 419 pixels) Return = random variable X. The distribution = continuous version of the histogram of X. Haijun Li Math/Stat 370: Engineering Statistics, Washington State University Week 2 12 / 23
30 Probability Density Function Probability Density Function (PDF) f (x) describes the probability density of a continuous RV X at x. 1 f (x) 0 2 IBUTIONS f (x)dx = 1 AND PROBABILITY DENSITY FUNCTIONS 3 P(a < X < b) = b f (x)dx = the area under f (x) between a a and b. f (x) P(a < X < b) a b x Haijun Li Math/Stat 370: Engineering Statistics, Washington State University Week 2 13 / 23
31 Probability Density Function PDF describes relative frequency of a continuous random variable X. Haijun Li Math/Stat 370: Engineering Statistics, Washington State University Week 2 14 / 23
32 Probability Density Function PDF describes relative frequency of a continuous random variable X. P(X = a) = 0. Haijun Li Math/Stat 370: Engineering Statistics, Washington State University Week 2 14 / 23
33 Probability Density Function PDF describes relative frequency of a continuous random variable X. P(X = a) = 0. P(a X b) = P(a < X b) = P(a X < b) = P(a < X < b). rb12_6.gif (GIF Image, pixels) Haijun Li Math/Stat 370: Engineering Statistics, Washington State University Week 2 14 / 23
34 Example The PDF of the failure time X of an electronic component in a copier (in hours) is f (x) = 1 [ 3000 exp x ], x 0, 3000 and zero otherwise. Find the probability that 1 A component lasts more than 1000 hours before failure. Haijun Li Math/Stat 370: Engineering Statistics, Washington State University Week 2 15 / 23
35 Example The PDF of the failure time X of an electronic component in a copier (in hours) is f (x) = 1 [ 3000 exp x ], x 0, 3000 and zero otherwise. Find the probability that 1 A component lasts more than 1000 hours before failure. P(X > 1000) = e x 3000 dx = [ e x 3000 ] 1000 = Haijun Li Math/Stat 370: Engineering Statistics, Washington State University Week 2 15 / 23
36 Example The PDF of the failure time X of an electronic component in a copier (in hours) is f (x) = 1 [ 3000 exp x ], x 0, 3000 and zero otherwise. Find the probability that 1 A component lasts more than 1000 hours before failure. P(X > 1000) = e x 3000 dx = [ e x 3000 ] 1000 = A component fails between 1000 and 2000 hours. Haijun Li Math/Stat 370: Engineering Statistics, Washington State University Week 2 15 / 23
37 Example The PDF of the failure time X of an electronic component in a copier (in hours) is f (x) = 1 [ 3000 exp x ], x 0, 3000 and zero otherwise. Find the probability that 1 A component lasts more than 1000 hours before failure. P(X > 1000) = e x 3000 dx = [ e x 3000 ] 1000 = A component fails between 1000 and 2000 hours. P(1000 < X < 2000) = e x 3000 dx = [ e x 3000 ] = Haijun Li Math/Stat 370: Engineering Statistics, Washington State University Week 2 15 / 23
38 Example The PDF of the failure time X of an electronic component in a copier (in hours) is f (x) = 1 [ 3000 exp x ], x 0, 3000 and zero otherwise. Find the probability that 1 A component lasts more than 1000 hours before failure. P(X > 1000) = e x 3000 dx = [ e x 3000 ] 1000 = A component fails between 1000 and 2000 hours. P(1000 < X < 2000) = e x 3000 dx = [ e x 3000 ] = Determine the number of hours at which 10% of all components have failed. Haijun Li Math/Stat 370: Engineering Statistics, Washington State University Week 2 15 / 23
39 Example The PDF of the failure time X of an electronic component in a copier (in hours) is f (x) = 1 [ 3000 exp x ], x 0, 3000 and zero otherwise. Find the probability that 1 A component lasts more than 1000 hours before failure. P(X > 1000) = e x 3000 dx = [ e x 3000 ] 1000 = A component fails between 1000 and 2000 hours. P(1000 < X < 2000) = e x 3000 dx = [ e x 3000 ] = Determine the number of hours at which 10% of all components have failed. Find a such that P(X a) = 0.1. That is, 1 e a 3000 = 0.1. Solve this, we have a 316 hours. Haijun Li Math/Stat 370: Engineering Statistics, Washington State University Week 2 15 / 23
40 Cumulative Distribution Function = Cumulative Frequency Cumulative Distribution Function (CDF) of X F(x) = P(X x) = x f (u)du. Haijun Li Math/Stat 370: Engineering Statistics, Washington State University Week 2 16 / 23
41 Cumulative Distribution Function = Cumulative Frequency Cumulative Distribution Function (CDF) of X F(x) = P(X x) = 1 F(x) is non-decreasing. 2 0 F(x) 1. 3 F(x) 1 as x +. x f (u)du. Haijun Li Math/Stat 370: Engineering Statistics, Washington State University Week 2 16 / 23
42 Figure: Comparison of PDF and CDF Haijun Li Math/Stat 370: Engineering Statistics, Washington State University Week 2 17 / 23
43 1. P(a X b) = F (b) F(a) = b a f (x)dx 2. f (x) = F (x).gif (GIF Image, pixels) Haijun Li Math/Stat 370: Engineering Statistics, Washington State University Week 2 18 / 23
44 Example Consider a PDF f (x) = 1 [ 3000 exp x ], x 0, 3000 and zero otherwise. Find the CDF. Haijun Li Math/Stat 370: Engineering Statistics, Washington State University Week 2 19 / 23
45 Example Consider a PDF f (x) = 1 [ 3000 exp x ], x 0, 3000 and zero otherwise. Find the CDF. F(x) = x 0 1 u 3000 e 3000 du = 1 e x 3000, x 0. Haijun Li Math/Stat 370: Engineering Statistics, Washington State University Week 2 19 / 23
46 Example Consider the PDF of the uniform RV over [a, b]. f (x) = 1 b a, a x b, and zero otherwise. Find the CDF. Haijun Li Math/Stat 370: Engineering Statistics, Washington State University Week 2 20 / 23
47 Example Consider the PDF of the uniform RV over [a, b]. f (x) = 1 b a, a x b, and zero otherwise. Find the CDF. F(x) = x 1 x a du =, a x b. a b a b a F(x) = 0, x < a. F(x) = 1, b < x. Haijun Li Math/Stat 370: Engineering Statistics, Washington State University Week 2 20 / 23
48 Mean and Variance Mean of a continuous RV X E(X) = µ = xf (x)dx = x f (x)dx. }{{} prob. mass at x Haijun Li Math/Stat 370: Engineering Statistics, Washington State University Week 2 21 / 23
49 Mean and Variance Mean of a continuous RV X E(X) = µ = xf (x)dx = Variance of a continuous RV X V (X) = σ 2 = (x µ) 2 f (x)dx = x f (x)dx. }{{} prob. mass at x (x µ) 2 f (x)dx }{{} prob. mass at x Haijun Li Math/Stat 370: Engineering Statistics, Washington State University Week 2 21 / 23
50 Mean and Variance Mean of a continuous RV X E(X) = µ = xf (x)dx = Variance of a continuous RV X V (X) = σ 2 = (x µ) 2 f (x)dx = x f (x)dx. }{{} prob. mass at x (x µ) 2 f (x)dx }{{} prob. mass at x Standard deviation (SD) of X: σ = V (X). Haijun Li Math/Stat 370: Engineering Statistics, Washington State University Week 2 21 / 23
51 Mean and Variance Mean of a continuous RV X E(X) = µ = xf (x)dx = Variance of a continuous RV X V (X) = σ 2 = (x µ) 2 f (x)dx = x f (x)dx. }{{} prob. mass at x (x µ) 2 f (x)dx }{{} prob. mass at x Standard deviation (SD) of X: σ = V (X). σ 2 = x 2 f (x)dx µ 2. Haijun Li Math/Stat 370: Engineering Statistics, Washington State University Week 2 21 / 23
52 Figure: σ y = SD of Y. Haijun Li Math/Stat 370: Engineering Statistics, Washington State University Week 2 22 / 23
53 Example Suppose that the CDF of the length (in millimeters) of computer cables is 0 x 1200 F(x) = 0.1x < x x > Find P(X < 1208), and P(1195 < X < 1205). Haijun Li Math/Stat 370: Engineering Statistics, Washington State University Week 2 23 / 23
54 Example Suppose that the CDF of the length (in millimeters) of computer cables is 0 x 1200 F(x) = 0.1x < x x > Find P(X < 1208), and P(1195 < X < 1205). P(X < 1208) = F(1208) = 0.8 P(1195 < X < 1205) = F(1205) F(1195) = F(1205) = 0.5. Haijun Li Math/Stat 370: Engineering Statistics, Washington State University Week 2 23 / 23
55 Example Suppose that the CDF of the length (in millimeters) of computer cables is 0 x 1200 F(x) = 0.1x < x x > Find P(X < 1208), and P(1195 < X < 1205). P(X < 1208) = F(1208) = 0.8 P(1195 < X < 1205) = F(1205) F(1195) = F(1205) = What is its PDF and mean? Haijun Li Math/Stat 370: Engineering Statistics, Washington State University Week 2 23 / 23
56 Example Suppose that the CDF of the length (in millimeters) of computer cables is 0 x 1200 F(x) = 0.1x < x x > Find P(X < 1208), and P(1195 < X < 1205). P(X < 1208) = F(1208) = 0.8 P(1195 < X < 1205) = F(1205) F(1195) = F(1205) = What is its PDF and mean? f (x) = F (x) = 0.1, 1200 < x E(X) = xdx = [0.05x 2 ] = Haijun Li Math/Stat 370: Engineering Statistics, Washington State University Week 2 23 / 23
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