Question

If $m$ and $\sigma^{2}$ are the mean and variance of the random variable $X$, whose distribution is given by

$X=x$	$0$	$1$	$2$	$3$
$P(X=x)$	$\frac{1}{3}$	$\frac{1}{2}$	$0$	$\frac{1}{6}$

Then

• A. $ m=\sigma ^{2}=2 $
• B. $ m=1,\sigma ^{2}=2 $
• C. $ m=\sigma ^{2}=1 $
• D. $ m=2,\sigma ^{2}=1 $

Answer 1

Answer: (C)

Given, distribution is

$X=x$	$0$	$1$	$2$	$3$
$P(X=x)$	$\frac{1}{3}$	$\frac{1}{2}$	$0$	$\frac{1}{6}$

$\therefore $ Mean, $m=\displaystyle\sum_{i=1}^{4} p_{i} x_{i}$
$=0 \times \frac{1}{3}+1 \times \frac{1}{2}+2 \times 0+3 \times \frac{1}{6}$
$=0+\frac{1}{2}+0+\frac{1}{2}=1$
Variance, $\sigma^{2}=\displaystyle\sum_{i=1}^{4} p_{i}\left(x_{i}-m\right)^{2}$
$=\frac{1}{3}(0-1)^{2}+\frac{1}{2}(1-1)^{2}+0(2-1)^{2}+\frac{1}{6}(3-1)^{2}$
$=\frac{1}{3}+0+0+\frac{2}{3}=1$
$\therefore m=\sigma^{2}=1$