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 |
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*}
\right. \quad \forall x=0,1,2,\dots \\
;p=\frac{3}{4} , p+q=1\begin{align*}
E(X) &= Pr[E=1]*E(X/E=1) + Pr[E=0]*E(X/E=0)
\end{align*}\begin{align*}
E(X/E=1) &= \sum_{x=0}^{\infty} x pq^x \\
&= pq \sum_{x=1}^{\infty} x q^{x-1} \\
&= \frac{pq}{p^2} = \frac{q}{p} \\
E(X/E=0) &= \frac{p}{q}
\end{align*}2E(X) = 2(q+p)=2
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