Outline Discrete Probability Distributions The Binomial - PDF document

12/7/2006 219323 Probability y and Statistics for Software and Knowledge Engineers Lecture 4: Discrete and Continuous Probability Distributions Monchai Sopitkamon, Ph.D. Outline � Discrete Probability Distributions – The Binomial Distribution (3.1) – The Geometric and Negative Binomial Th G t i d N ti Bi i l Distributions (3.2) – The Hypergeometric Distribution (3.3) – The Poisson Distribution (3.4) – The Multinomial Distribution (3.5) � Continuous Probability Distributions – The Uniform Distribution (4.1) – The Exponential Distribution (4.2) – The Gamma Distribution (4.3) – The Weibull Distribution (4.4) – The Beta Distribution (4.5) 1

12/7/2006 Discrete Probability Distributions: The Binomial Distribution I (3.1) � Bernoulli RVs – Discrete RVs that take just two values – Can be used to model the outcome of a coin toss, whether an item is defective or not, whether a student will pass or fail this course, and etc. – Outcomes are labeled 0 and 1 with parameter p (0 ≤ p ≤ 1) specifying the prob that the outcome is 1 that the outcome is 1 – E ( X ) = (0x P ( X =0))+ (1x P ( X =1)) = p – E ( X 2 ) = (0 2 x P ( X =0))+ (1 2 x P ( X =1)) = p – Var( X ) = E ( X 2 ) – ( E ( X )) 2 = p – p 2 = p (1 – p ) Discrete Probability Distributions: The Binomial Distribution II (3.1) � If n independent Bernoulli trials are performed, each with a constant prob p of “ success” , then the RV X = X 1 +…+ f “ h h RV X ” X X n has a binomial distribution with parameters n and p � X ∼ B ( n , p ) counts the number of successes in the n trials ⎛ ⎞ � Pmf of a B ( n , p ) RV: ( , p ) n = = = = ⎜ ⎜ ⎟ ⎟ − − x n x ( ( ) ) ( ( 1 1 ) ) P P X X x x ⎜ ⎜ ⎟ ⎟ p p p p ⎝ ⎠ x � E ( X ) = E ( X 1 )+…+ E ( X n ) = p +…+ p = np � Var( X ) = Var ( X 1 ) +…+ Var( X n ) = p (1- p ) +…+ p (1- p ) = np (1- p ) 2

12/7/2006 Discrete Probability Distributions: The Binomial Distribution III (3.1) X ∼ B (4 , p ) Discrete Probability Distributions: The Binomial Distribution IV (3.1) Symmetric Prob Distribution P ( X ≥ 6) = 1- P ( X ≤ 5) B ( n , 0.5) with respected to E ( X )= n /2 = 1-0.855 = 0.145 Figure 3.2 Probability mass function and cumulative distribution function of a B (8, 0.5) random variable 3

12/7/2006 Discrete Probability Distributions: The Binomial Distribution V (3.1) ⎛ ⎞ ⎛ ⎞ 3 ⎛ − ⎞ 5 ⎛ ⎞ 3 ⎛ ⎞ 5 8 1 1 8 ! 1 2 = = ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ = ⎜ ⎟ ⎜ ⎟ P ( X 3 ) 1 ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ 3 3 3 3 ! 5 ! 3 3 = 0.273 P ( X ≤ 1) = P ( X =0) + P ( X =1) = 0.039 + 0.156 = 0.195 Figure 3.3 Probability mass function and cumulative distribution function of a B (8, 1 / 3) random variable Discrete Probability Distributions: The Binomial Distribution VI (3.1) P ( X =4) = 0.171 P ( X ≤ 2) = P ( X =0) + P ( X =1) + P ( X =2) = 0 001 + 0 002 + 0 017 0.001 + 0.002 + 0.017 = 0.020 Figure 3.4 Probability mass function and cumulative distribution function of a B (8, 2 / 3) random variable 4

12/7/2006 Discrete Probability Distributions: The Binomial Distribution VII (3.1) Ex. If the probability of a student successfully passing this course (C or better) is 0.82, find the passing this course (C or better) is 0.82, find the probability that given 8 students ⎛ ⎞ 8 ( ) ( ) 8 0 ≈ ⎜ ⎟ a. all 8 pass. 0.82 0.18 0.2044 ⎝ 8 ⎠ ⎛ ⎞ 8 ( ) ( ) b. none pass. 0 8 ≈ ⎜ ⎟ 0.82 0.18 0.0000011 ⎝ ⎠ 0 c. at least 6 pass. ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 8 8 8 ( ) ( ) ( ) ( ) ( ) ( ) 6 2 + 7 1 + 8 0 ⎜ ⎟ 0.82 0.18 ⎜ ⎟ 0.82 0.18 ⎜ ⎟ 0.82 0.18 ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ 6 7 8 ≈ + + = 0.8392 0.2758 0.3590 0.2044 Outline � Discrete Probability Distributions – The Binomial Distribution (3.1) – The Geometric and Negative Binomial Th G t i d N ti Bi i l Distributions (3.2) – The Hypergeometric Distribution (3.3) – The Poisson Distribution (3.4) – The Multinomial Distribution (3.5) � Continuous Probability Distributions – The Uniform Distribution (4.1) – The Exponential Distribution (4.2) – The Gamma Distribution (4.3) – The Weibull Distribution (4.4) – The Beta Distribution (4.5) 5

12/7/2006 Discrete Probability Distributions: The Geometric Distribution I (3.2.1) � The number of trials up to and including the first success in a sequence of the first success in a sequence of independent Bernoulli trials with a constant success prob p has a geometric dist w/ parameter p � The pmf P ( X = x ) = (1 – p ) x -1 p for x = 1, = = − − x 1 ( ) ( 1 ) P X x p p 2, 3, 4,… ≤ = − − x � The cdf X ≤ x ) = 1 – (1 – p ) ( ) 1 ( 1 ) P X x p − 1 ( 1 p ) = = ( ) E ( X ) Var X 2 p p Discrete Probability Distributions: The Geometric Distribution II (3.2.1) Prob getting head the first time after 3 failures = P ( X =4) = (1- p ) 4-1 p = (1/2) 3 x(1/2) = 1/16 3 Figure 3.11 Probability mass function and cumulative distribution function of a geometric distribution with parameter p = 1 / 2 6

12/7/2006 Discrete Probability Distributions: The Negative Binomial Distribution I (3.2.2) � The number of trials that gives r successes in a sequence of successes in a sequence of independent Bernoulli trials w/ a constant success prob p has a negative binomial dist w/ parameters p and r . ⎛ − ⎞ x 1 ( ( ) ) − ⎜ ⎜ ⎟ ⎟ � The pmf � The pmf = = = = − x r r ( ( ) ) 1 1 P P X X x x ⎜ ⎜ ⎟ ⎟ p p p p ⎝ − ⎠ 1 r for x = r , r + 1, r + 2, … − � E ( X ) = r / and r ( 1 p ) r = = ( ) Var ( X ) E X 2 p p Discrete Probability Distributions: The Negative Binomial Distribution II (3.2.2) Modeling number of tosses of a fair coin until until two heads are obtained. ⎛ ⎛ ⎞ ⎞ 1 1 = = ⎜ ⎜ ⎟ ⎟ × × = 0 0 2 2 P P ( ( X X 2 2 ) ) 0 0 . 5 5 0 0 . 5 5 ⎜ ⎟ ⎝ ⎠ 1 4 Figure 3.13 Probability mass function and cumulative distribution function of a negative binomial distribution with parameters p = ½ and r = 2 7

12/7/2006 Discrete Probability Distributions: The Negative Binomial Distribution III (3.2.2) � Ex. 12 pg.163: p = 0.6, r = 3 Prob that exactly six people P b th t tl i l need to be interviewed is ⎛ ⎞ 5 ⎜ ⎟ = = × × = 3 3 P ( X 6 ) ⎜ ⎟ 0 . 4 0 . 6 0 . 138 ⎝ ⎠ 2 The expected number of interviews required is interviews required is 3 r = p = = ( ) 5 E X 0 . 6 Discrete Probability Distributions: The Negative Binomial Distribution IV (3.2.2) Figure 3.16 Probability mass function of a negative binomial distribution with parameters p = 0 . 6 and r = 3, the distribution of the number of applicants interviewed 8

12/7/2006 Outline � Discrete Probability Distributions – The Binomial Distribution (3.1) – The Geometric and Negative Binomial Th G t i d N ti Bi i l Distributions (3.2) – The Hypergeometric Distribution (3.3) – The Poisson Distribution (3.4) – The Multinomial Distribution (3.5) � Continuous Probability Distributions – The Uniform Distribution (4.1) – The Exponential Distribution (4.2) – The Gamma Distribution (4.3) – The Weibull Distribution (4.4) – The Beta Distribution (4.5) Discrete Probability Distributions: The Hypergeometric Distribution I (3.3) � Used when selecting a group of identical r items out of a total of N objects, and � If n items are chosen at random without � If n items are chosen at random without replacement (prob of “success” is NOT constant), then � We have a hypergeometric distribution of the number of items of a certain kind in a random sample of size n drawn without replacement from a population of size N that contains r items of this kind. − ⎛ ⎛ ⎞ ⎞ ⎛ ⎛ ⎞ ⎞ r N r ⎜ ⎟ × ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ − ⎝ ⎠ ⎝ ⎠ x n x = = � The pmf is P ( X x ) ⎛ ⎞ N ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ n for max(0, n + r - N ) ≤ x ≤ min (n , r) 9

12/7/2006 Discrete Probability Distributions: The Hypergeometric Distribution II (3.3) � The expected value is The expected value is nr = E ( X ) N � The variance of ⎛ − ⎞ ⎛ − ⎞ N n r r = ⎜ ⎜ ⎟ ⎟ × × × × × × ⎜ ⎜ ⎟ ⎟ Var Var ( ( X X ) ) n n 1 1 − ⎝ ⎠ ⎝ ⎠ N 1 N N Discrete Probability Distributions: The Hypergeometric Distribution III (3.3) Ex.17 pg.168: N = 16 r = 6 n = 5 N = 16, r = 6, n = 5 ⎛ ⎞ ⎛ ⎞ 6 10 ⎛ ⎞ ⎛ ⎞ 6 ! 10 ! ⎜ ⎟ × ⎜ ⎟ × ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ 2 3 2 ! 4 ! 3 ! 7 ! = = = = P ( X 2 ) 0 . 412 ⎛ ⎞ ⎛ ⎞ 16 16 ! ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ 5 ! 11 ! 5 The expected number of p underweight containers chosen by the inspector is × 5 6 nr = N = = E ( X ) 1 . 875 Figure 3 .1 7 Probability m ass 16 function of the num ber of underw eight m ilk containers in the inspector’s sam ple 10