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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