Question

A pair of fair dice is thrown. Let X be the random variable which denotes the minimum or equal of the two numbers which appear. Find the probability distribution, mean and variance of X.

Answer 1

Let $X$ denote the event of getting twice the number. Then, $X$ can take the values $1,2,3,4,5$ and $6$.

Thus, the probability distribution of $X$ is given by

$x$	$P\left(X\right)$
$1$	$\frac{11}{36}$
$2$	$\frac{9}{36}$
$3$	$\frac{7}{36}$
$4$	$\frac{5}{36}$
$5$	$\frac{3}{36}$
$6$	$\frac{1}{36}$

$x_i$	$p_i$	$p_i{x}_i$	$p_i{x}_i^2$
$1$	$\frac{11}{36}$	$\frac{11}{36}$	$\frac{11}{36}$
$2$	$\frac{9}{36}$	$\frac{18}{36}$	$1$
$3$	$\frac{7}{36}$	$\frac{21}{36}$	$\frac{63}{36}$
$4$	$\frac{5}{36}$	$\frac{20}{36}$	$\frac{80}{36}$
$5$	$\frac{3}{36}$	$\frac{15}{36}$	$\frac{75}{36}$
$6$	$\frac{1}{36}$	$\frac{6}{36}$	$1$
		$\sum p_i{x}_i=\frac{91}{36}=2.5$	$\sum p_i{x}_i^2=8.4$

$\Rightarrow$ Mean $=\sum p_i{x}_i=6.25$

$\Rightarrow$ Variance $=\sum p_i{x}_i^2-(Mean{)}^2$

$=8.4-6.25$

$=2.15$