Question

Problem 1. Stanislaw is collecting coupons. Each day he receives randomly one of n distinct coupons with equal probabilities (independently of other days (a) Let T be the number of days it takes Stanislaw to obtain a complete set. Explain why T can be written as a sum of n independent Geometric random variables and say what their parameters are (b) Compute the expected value of T. (Use the fact that the expectation of a sum of random variables is the sum of the expectations.)

Answer 1

$(a)~Let~X_i=random~variable~denoting~no.~of~days~required~to~get~ith\\ distinct~coupon,~i=1,2,3,...,n\\ Then~P(X_i=x)=\left(1-\frac{1}{n} \right )^{x-1}\times\left(\frac{1}{n} \right );~x=1,2,3,....\\ =0~otherwise\\ where~probability~of~getting~success~(getting~ith~distinct~coupon)=\frac{1}{n}=p,~say\\ Then~T=X_1+X_2+...+X_n\sim Negative~binomial\left(n,~p \right )\\ where,~p=\frac{1}{n}$

$The~pmf~of~T~is\\ P(T=t)=\binom{t-1}{t-n}p^n(1-p)^{t-n},~t=n,n+1,...\\ =0~otherwise.\\ (b)~E(X_i)=\sum_{x=1}^\infty xpq^{x-1},~q=1-p\\ =p\sum_{x=1}^\infty \frac{\mathrm{d} q^x}{\mathrm{d} q}=p\frac{\mathrm{d} }{\mathrm{d} q}\left(\sum_{x=1}^\infty q^x \right )=p\frac{\mathrm{d} }{\mathrm{d} q}\left(\frac{q}{1-q} \right )=p\times \frac{1}{(1-q)^2}=\frac{1}{p}=n\\ E(T)=\sum_{i=1}^nE(X_i)=n^2.$