A random variable $\mathrm{X}$ assumes value $1,2,3, \ldots \ldots . n$ with equal probabilities. If the…
A random variable $\mathrm{X}$ assumes value $1,2,3, \ldots \ldots . n$ with equal probabilities. If the ratio of variance of $=\sum p_i x_i^2-\left(\sum p_i x_i\right)^2$ to expected value of $\mathrm{X}$ is equal to 4 , then the value of $n$ is
35
50
30
25
Solution
$\begin{aligned} & \text { Variance }=\sum p_i X_i^2-\left(\sum p_i X_i\right)^2 \\ & =\frac{1}{n} \times \frac{n(n+1)(2 n+1)}{6}-\left\{\frac{1}{n} \times \frac{n(n+1)}{2}\right\}^2 \\ & =\frac{n(n+1)}{2 n}\left\{\frac{2 n+1}{3}-\frac{n(n+1)}{2 n}\right\} \\ & \text { Expected value }=\sum p_i x_i=\frac{1}{n} \cdot \frac{n(n+1)}{2}=\frac{n(n+1)}{2 n} \\ & \text { Ratio }=\frac{2 n+1}{3}-\frac{n(n+1)}{2 n}\end{aligned}$
$\Rightarrow n^2-25 n=0$
$\Rightarrow n(n-25)=0$
$\Rightarrow n=0$ or $n=25$
$\Rightarrow n=25$ as $n=0$ is not possible