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.

First Trial

Random Variable X

Distribution of X

Success

Number of failures before the first success.

Geometric

Failure

Number of successes before the first failure.

Geometric

Table 1:

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: