Q.4.8 - A First Course in Probability

Question

Find $Var\left(X\right)$ if $P(X = a) = p = 1 - P(X = b)$

Step-by-Step Solution

Answer

We have found to be$Var\left(X\right)=(a-b{)}^2(1-p)p$

Step 1: Given information

Given in the question is,

$P(X=a)=p=1-P(X=b)$

Step 2: Substitution

The random variable $X$ can consider only two values, $a$ with probability $p$ and $b$ with the probability $1-p$.

Using the meaning of the mean, we have that

$E\left(X\right)=a\times p+b\times (1-p)=:\mu$

Step 3: Calculation

Currently, operating the definition of the variance, we have that

$Var\left(X\right)=E\left((X-\mu {)}^2\right)=(a-\mu {)}^2·p+(b-\mu {)}^2·(1-p)$

$=(a-a\times p-b\times (1-p){)}^{2}p+(b-a\times p+b\times (1-p){)}^2(1-p)$

Substitute the given expression,

$=(a-b{)}^2(1-p{)}^{2}p+(b-a{)}^2{p}^2(1-p)$

$=(a-b{)}^2(1-p\left)p\right[1-p+p]$

We get,

$=(a-b{)}^2(1-p)p$.

Step 4: Final answer

The solution of the $Var\left(X\right)$ is found to be $(a-b{)}^2(1-p)p$.