Week 4 notes

Question 1

How is a hypergeometric distribution defined?

Answer 1

Suppose in a collection of N objects, m are of type 1 and N-m are of another type 2. Furthermore, suppose that n objects are randomly selected from the collection without replacement. Define the random variable X to give the number of selected objects that are of type 1. THen X has a hyper geometric distribution with parameters N, m, n. The probability mass function of X is given by:
p(x) =P(X = x)
= (m choose x)*( N-m choose n-x)/(N choose n)

Question 2

How is a geometric distribution defined?

Answer 2

Suppose that a sequence of independent Bernoulli trials is performed, with p = P(“success”) for each trial. Define the random variable X to five the number of the trial at which the first success occurs. Then X has a geometric distribution with parameter p. The probability mass function of X is given by:
p(x) = P(X = x) = ((1-p)^(x-1))*p

Question 3

How is a negative binomial distribution defined?

Answer 3

Suppose that a sequence of independent Bernoulli trials is performed, with p = P(“success”) for each trial. FIx an integer r to be greater than or equal to 2 and define the random variable X to give the number of the trial at which the rth success occurs. THen X has a negative binomial distribution with parameters r and p. The probability mass function of X is given by:
p(x) = P(X=x) =(x-1 choose r-1)(p^r)((1-p)^(x-r))

Question 4

How is a Poisson distribution defined?

Answer 4

A random variable X has a Poisson distribution if its frequency function is given by:
p(x) = P(X=x) = ((e^(-gamma))*(gamma^x))/x!,
for x is a non-negative integer where gamma greater than zero is a paramter of the distrubuion. We write X~Poisson(gamma).