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DistributionSummary.doc

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Department
Statistics
Course
STAT 230
Professor
Diana Parry
Semester
Winter

Description
Summary of Distributions in the Text Book 1 t-distribution Z ~ Std. Normal (0,1) ν→∞ Z = Y − μY σY 2 Y =lnX 2 Hypergeometric (N,k,n) Y ~ Normal (μY,Y ) X ~ Log-normal (μX,σX) and independent trials p not close to 0,1 2 (replacement) 2 λt →∞ μ =σY=λt Y ) (μY= np, σY=np(1-p)) Binomial (n,p) Poisson (λt) n→∞,p→0 λt=np ( ) BERNOULLI POISSON PROCESS PROCESS Beta2 Negative Binomial (k,p) Gamma Chi-squared 3 special case special cases Geometric (1,p) Exponential (λ) Note: You are only responsible for those distributions that are covered in lecture (not beta or gamma) 1 The continuous uniform and F distributions are not illustrated because they are somewhat unrelated to these distributions. The multinomial, multivariate hypergeometric, bivariate normal, and Weibull distributions are not shown and are not covered in lecture. 2 The beta distribution relates to the probability of success in a set of Bernoulli trials (e.g. estimating p for an unfair coin). 3 The gamma distribution has a number of special cases, one of which (the Erlang distribution) is the waiting time until the n event th in a Poisson process. The exponential distribution is the waiting time until the first such event. BINOMIAL DISTRIBUTION A Bernoulli trial can result in a success with probability p or a failure with probability q = 1 – p. If X is the binomial random variable for the number of successes in n independent trials, then the probability distribution of X is: x n-x P(X = x)= C n x1− p) , x = 0,1,2,..., n E[X ] = np , Var[X ] = np(1− p) HYPERGEOMETRIC DISTRIBUTION If X is the hypergeometric random variable for the number of successes in a random sample of size n selected from N items of which k are successes and N – k are failures, then the probability distribution of X is: P(X = x)= C k x N-k n-x N n N − n nk k E[X ] = nk/N , Var[X ] = . (1− ) N −1 N N NEGATIVE BINOMIAL DISTRIBUTION If repeated independent trials can result in a success with probability p or a failure with probability q = 1 – p, then the probability distribution of the random variable X, the number of th the trial on which the k success occur
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