Question

A coin, having probability \(p\) of landing heads, is flipped until a head appears for the \(r\) th time. Let \(N\) denote the number of flips required. Calculate \(E[N]\). Hint: There is an easy way of doing this. It involves writing \(N\) as the sum of \(r\) geometric random variables.

Short answer

The expected value of the number of flips required to get the $r$-th head with probability $p$ can be calculated by representing $N$ as the sum of $r$ geometric random variables and using the linearity of expectation. This gives us \(E[N] = \frac{r}{p}\).

Step-by-step solution

Understand the concept of geometric random variables

A geometric random variable is defined as the number of trials needed to get the first success in a sequence of Bernoulli trials. In this problem, a success is defined as getting a head, and the probability of success in each flip is p.

Represent N as the sum of r geometric random variables

Let Xi be the number of flips required to get the i-th head, for i = 1, 2, ..., r. Since each head is an independent event, the total number of flips (N) required to get r heads can be represented as the sum of the number of flips needed for each individual head: \[N = X_1 + X_2 + ... + X_r\]

Determine the expected value of each geometric random variable

The expected value of a geometric random variable with success probability p is given by the formula: \(E[X] = \frac{1}{p}\). Since each Xi is geometric with probability p, the expected value of each Xi is also \(\frac{1}{p}\).

Calculate the expected value of N using the linearity of expectation

The linearity of expectation states that the expected value of a sum of random variables is equal to the sum of the expected values of each random variable. As such, we can write the expected value of N as follows: \[E[N] = E[X_1] + E[X_2] + ... + E[X_r]\] Since the expected value of each Xi is \(\frac{1}{p}\), we can rewrite the expression as: \[E[N] = r \times \frac{1}{p} = \frac{r}{p}.\] So, the expected value of N, the number of flips required to get r heads, is \(\frac{r}{p}\).