Let $n$ different types of coupons. Jhon wishes to collect them all. Each day he receives one coupon with uniform distribution with complete independence between the days. Let $X$ be a random variable that marks the number of days until Jhon collected all coupons. Find the expected value of $X$.
I have written some proof, but there is one part where I'm not sure I'm executing the correct calculations. I'd love someone else to have a look.
First let $X_i$ be a random variable marking the number of days it took Jhon to collect the $i$ coupon after being the owner of $i-1$ coupons. Denote, $X = X_1+...+X_n$.
From the linear property we have: $$E[X] = \sum_{i=1}^{n} E[X_i]$$
This is the part I'm unsure about:
Let us compute $E[X_i]$. Denote, $E[X_i]$ is the average number of days it took Jhon to draw one of $n-i+1$ coupon whilst receiving each day one of $n$ coupons with uniform distribution. If we think Jhon would be satisfied with each on the $n$ coupons then we would have $E[X_i] = {n \over 2}$ so since in this case we have ($n= n-i+1$) then $E[X_i] = {n-i+1 \over 2}$.
This leaves us with: $$E[X] = \sum_{i=1}^n {n-i+1 \over 2} = {1 \over 2}\sum_{i=1}^n(n+1)-{1 \over 2}\sum_{i=1}^ni=...={n(2n+3) \over 4}$$
Thank you in advance!
