Question

Three numbers are selected at random (without replacement) from first six positive integers. Let $X$ denote the largest of the three numbers obtained. Find the probability distribution of $X$. Also, find the mean and variance of the distribution.

Answer 1

$x$	$p\left(x\right)$	$xp\left(x\right)$	$x^{2}p\left(x\right)$

First six positive integers are $1,2,3,4,5,6.$

Therefore, Three numbers are selected at random without replacement so, total no. of ways ${}^6{C}_3=20$.

Let, $x$ denote the larger of three numbers.

So, $x$ can take values $3,4,5,6.$

$p(x=3)=\frac{1}{20}$

$p(x=4)=\frac{3}{20}$

$p(x=5)=\frac{6}{20}$

$p(x=6)=\frac{10}{20}$

Refer the tabular column:

$mean=\sum xp\left(x\right)$

$=\frac{3}{20}+\frac{12}{20}+\frac{30}{20}+\frac{60}{20}$

$=\frac{150}{20}=5.25$

Variance $=\sum x^{2}p\left(x\right)-{(\sum xp(x\left)\right)}^2$

$=\left(\frac{567}{20}\right)-{\left(\frac{105}{20}\right)}^2$

$=28.35-27.56=\mathrm{0.79.}$