Question

A fair die is tossed. Let X denote twice the number appearing. 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 2, 4, 6, 8, 10 and 12.

Thus, the probability distribution of X is given by

X	P(X)
2	$\frac{1}{6}$
4	$\frac{1}{6}$
6	$\frac{1}{6}$
8	$\frac{1}{6}$
10	$\frac{1}{6}$
12	$\frac{1}{6}$

Computation of mean and variance

xi	pi	pixi	pixi2
2	$\frac{1}{6}$	$\frac{2}{6}$	$\frac{4}{6}$
4	$\frac{1}{6}$	$\frac{4}{6}$	$\frac{16}{6}$
6	$\frac{1}{6}$	$\frac{6}{6}$	$\frac{36}{6}$
8	$\frac{1}{6}$	$\frac{8}{6}$	$\frac{64}{6}$
10	$\frac{1}{6}$	$\frac{10}{6}$	$\frac{100}{6}$
12	$\frac{1}{6}$	$\frac{12}{6}$	$\frac{144}{6}$
		${\sum }_{}^{}$pixi = 7	${\sum }_{}^{}$pixi2 = $\frac{364}{6}$

$$\begin{gathered} \mathrm{Mean}=\sum p_i{x}_i=7 \\ \mathrm{Variance}=\underset{}{\overset{}{\sum p_i{x_i}^2}}-{\left(\mathrm{Mean}\right)}^2 \\ =60.7-49 \\ =11.7 \end{gathered}$$