# JAM 2022 [ 41 -50]

Consider a sequence of independent Bernoulli trials, where 3/4 is the probability of success in each trial. Let X be a random variable defined as follows: If the first trial is a success, then X counts the number of failures before the next success. If the first trial is a failure, then X counts the number of success before the next failure. Then 2E(X) equals ______________________.

Since we are given a sequence of independent Bernoulli trials. Each trial is independent of the other. Hence, we can say the following about the random variable X.

Suppose, E is another random variable which takes values 1 or 0 depending upon whether the first trial is a success or failure respectively. Then the pmf of the random variable X may be given as:

f(x) = \left\{
\begin{align*}
pq^{x} ; & \text{if E=1}\\
qp^{x} ; & \text{if E=0}
\end{align*}