Question

Your employer wants to show you their appreciation, so for Easter they get you a one year membership in the Jelly of the Month Club (it's the gift that keeps on giving the whole year). Each month you are sent one of n different flavours of jelly uniformly at random, and independently of previous orders. Define the random variable X to be the total number of distinct brands of jelly that you receive. Determine the expected value E(X) of X. Hint: Use indicator random variables.

Answer 1

The expected value of the total number of distinct brands of jelly that you will receive in the Jelly of the Month Club is 1.

Step-by-step explanation:

The expected value, E(X), of the random variable X in this scenario can be determined using indicator random variables. An indicator random variable is a binary random variable that takes on the value 1 when a certain event occurs and 0 otherwise. In this case, we can define n indicator random variables, one for each brand of jelly. Let Xi be the indicator random variable that takes on the value 1 if the i-th brand of jelly is received and 0 otherwise.

The probability that Xi = 1 is the probability of receiving the i-th brand of jelly in a given month, which is 1/n. Therefore, the expected value of each indicator random variable is E(Xi) = 1/n.

The total number of distinct brands of jelly that you receive, X, is the sum of the indicator random variables Xi. Therefore, the expected value of X can be calculated as:

E(X) = E(X1 + X2 + ... + Xn)

E(X) = E(X1) + E(X2) + ... + E(Xn)

E(X) = 1/n + 1/n + ... + 1/n

E(X) = n * (1/n)

E(X) = 1