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STA3032 Chapter 4

# STA3032 Chapter 4 - Revised Chapter 4 Some Commonly Used...

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Revised 9/8/2010 Chapter 4 Some Commonly Used Distributions Sections 4.1 to 4.4 Distribution of Discrete Random Variables The Binomial Family of distributions The binomial family includes the Bernoulli distribution, Binomial distribution, Geometric distribution, Negative Binomial distribution, Poisson distribution and the Multinomial Distribution. In all but the last one, we assume that there is an experiment with two possible outcomes, labeled “Success” (or S = the outcome we are interested in and count) and “Failure” (or F) You should notice that these distributions are all inter-related. There are similarities and differences between them. The differences define when you should use each one. Here is a list of these distributions with their similarities and differences: Distributions and Conditions when they are used: In each of the following distributions, we are assuming that a Bernoulli experiment (with only two possible outcomes, S and F) is repeated. Distributions Conditions Number of Repetitions Number of “Success”s P(“Success”) Repetitions independent? Page B(n,p) n (fixed) Random Variable π Yes 202 G(p) Random Variable One (fixed) π Yes 232 nb(r,p) Random Variable r (fixed) π Yes 235 P(λ) Infinity Random Variable π = λt Yes 214 H(N, R, n) n (fixed) Random Variable Changes No 229 STA3032, Chapter 4, Page 1 of 11

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Properties of the Distributions As before, we are assuming that a Bernoulli experiment (with only two possible outcomes, S and F) is repeated in each of the following cases. Distributions Properties p(x) =P(X = x) [ 1 ] μ =mean σ 2 = variance Page Bernoulli 1 (1 ) x x π π - - for x = 0, 1 π π×(1 – π) 199 Binomial B(n,p) (1 ) x n x n x π π - �� - �� �� for x = 0, 1, 2, …, n n×π n×π×(1-π) 202 Geometric G(p) 1 (1 ) x π π - - for x = 1, 2, … 1/π (1 – π)/π 2 232 Negative Binomial nb(r,p) 1 (1 ) 1 r x r x r π π - - - - for x = r, r+1, r+2, … r/π r×(1 – π)/π 2 235 Poisson P(λ) / ! x e x λ λ - for x = 0, 1, 2, … λ λ 214 Hyper- geometric H(N, R, n) / R N R N x n x n - � �� �� � � �� �� � - � �� �� � for x = 0, 1, …, n ( 2 ) n×π, where π = R/N ( 3 ) ( (1 ) 1 N n n N π π - ° - - 229 1 Note that p(x) = 0 for values of x not specified in this column. 2 This assumes we define 0 a b �� = �� �� whenever a < b for any integers a, b. 3 Here we have π = P(“Success”) on the first draw . STA3032, Chapter 4, Page 2 of 11
The multinomial distribution: This is an extension of the binomial distribution where the experiment has k possible outcomes, each probability π i , i = 1, 2, …, k such that 1 1. k i i π = = The experiment is repeated n times and the number of times each outcome is observed (X i ) is the random variable of interest. Again, we have the constraint that 1 . k i i x n = = Under these conditions, the joint probability distribution of X 1 , X 2 , …, X k is 1 2 1 2 1 1 2 2 1 2 1 1 2 ( , , ..., ) ( , , ..., ) ... 0,1,2,..., , , ..., 0 k k k k k x x x k i i i k p x x x P X x X x X x n x n and x n x x x otherwise π π π = = = = = = = = ± ± ± This seems to be a good place to solve some problems from the supplementary exercises on page 318. Try problems 1, 3, 5, 14, 20,

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STA3032 Chapter 4 - Revised Chapter 4 Some Commonly Used...

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