Learn / Library / Expected value

Random number of coins

Expected valueHard

A fair $4$ -sided die is rolled to give a number $N$ from $1$ to $4$ . Then $N$ fair coins are flipped, and the payout is the number of heads. What is the variance of the payout?

Checking an answer needs JavaScript. The hint and the solution open without it.

Hint

Condition on the die result and split the variance accordingly.

Solution

By the law of total variance, $Var (X) = E [Var (X ∣ N)] + Var (E [X ∣ N])$ .
Given $N$ , the head count has variance $\frac{N}{4}$ , so the first term is $\frac{E [ N ]}{4} = \frac{5}{8}$ .
Given $N$ the mean is $\frac{N}{2}$ , so the second term is $\frac{Var ( N )}{4} = \frac{\frac{5}{4}}{4} = \frac{5}{16}$ .
The total is $\frac{5}{8} + \frac{5}{16} = \frac{15}{16}$ .

Answer: $\frac{15}{16}$

Next question Back to the library