Question

a. Suppose that A and B are two independent integer-valued random variables with P(A=k)=(n1 choose k)p^k(1-p)^n1-k and P(B=k)=(n2 choose k)p^k(1-p)^n2-k for some positive integers n1 and n2. Write (and simplify) a formula for P(A=k | A+B=m). For which values of k is this probability nonzero?

b. A sample of m balls is drawn (without replacement) from an urn containing n1 white balls and n2 black balls. Let W be the number of white balls in the sample. Write a formula for P(W=k).

c. Your answer for part a and b should be the same. Explain why this is.(Hint: Interpret the variables A and B in the context of coin flips, and interpret the conditional probability computed in part a in terms of those coin flips.)

Answer 1

W follows hypergeometric distribution with parameter{ (n1+n2),n2,m}

n1+n2 range {1,2.....}

n2 range {1,2.....}

m range {1,2.....( n1+n2)}

(C). Answer for part a and part b are same.

Let variable A indicates flip of a coin with event n1 and B also indicates flip of another coin with event n2. Therefore (n1+n2) be the sum of events from where m sample drawn. If there are k sample is drawn from n1 events then (m-k) sample is drawn from n2 events. It is a conditional part which is similar to part b. We know that 'with replacement' sample follows binomial distribution and 'without replacement ' follows hypergeometric distribution of same sample.... And the conditional distribution of binomial distribution of two variable follows hypergeometric distribution. Thats way the answer of part a and part b are same.