Bud sells $n$ different brands of ales. When you place an order, Bud sends you one bottle of ale, chosen uniformly at random from the $n$ different brands, independently of previous orders.
Jim wants to try all different brands of ale. He repeatedly places orders at Bud (one bottle per order) until he has received at least one bottle of each brand.
Define the random variable $X$ to be the total number of orders that
Jim places.
Determine the expected value $E(X)$. Use Linearity of Expectation. If Jim has received exactly
$i$ different brands of ale, how many orders does he expect to place
until he receives a new brand?
I'm unsure of how to use Linearity of Expectation to solve this problem. I have the following so far.
$x$ = the total number of orders that Jim places.
Since we know with every new brand of ale Jim receives, then his chance to get a new ale is less.
$E(x) = \sum_{i=0}^n (1-\frac{i}{n})$
First ale he receives, probability of a new type is and as follows: $1, 1-\frac{1}{n}, 1-\frac{2}{n},...,1-\frac{n-1}{n},1-\frac{n}{n}$ until he gets n different ales then there is no chance for him to receives new brands.
Could someone assist me with this proof?
