From an urn, which contains a white and b black balls,we select n balls with replacement.

However, each time a ball is put back into the urn, we also put s more balls of the same color into the urn. Let X be the number of white balls in the sample of n balls selected.

(i) Show that the probability function of X is given by the formula

The distribution of the random variable X is often referred to as the Pólya distribution, and is in fact one of several distributions named after the Hungarian mathematician George Pólya (1887–1985).
(ii) Which distribution arises in the special case s = 0?
(iii) Suppose now that s = −1 so that, rather than putting more balls, we remove one ball from the urn, which is of the same color as the last ball chosen. What is the distribution of X in this case?

f(x) = P(X = x) = (") + i=1 n-x (a+(i-1)s) (b+ (i 1)s) i=1 (a+b+(i-1)s) El