Question

A fair die is tossed. Let X denote 1 or 3 according as an odd or an even number appears. Find the probability distribution, mean and variance of X.

Answer 1

Let X be 1 for the appearance of odd numbers 1, 3 or 5 on the die. Then,
$P\left(X=1\right)=\frac{3}{6}=\frac{1}{2}$

Let X be 3 for the appearance of even numbers 2, 4 or 6 on the die. Then,
$P\left(X=3\right)=\frac{3}{6}=\frac{1}{2}$

Thus, the probability distribution of X is given by

X	P(X)
1	$\frac{1}{2}$
3	$\frac{1}{2}$

Computation of mean and variance

xi	pi	pixi	pixi2
1	$\frac{1}{2}$	$\frac{1}{2}$	$\frac{1}{2}$
3	$\frac{1}{2}$	$\frac{3}{2}$	$\frac{9}{2}$
		${\sum }_{}^{}$pixi = 2	${\sum }_{}^{}$pixi2 = 5

$$\begin{gathered} \mathrm{Mean}=\sum p_i{x}_i=2 \\ \mathrm{Variance}=\underset{}{\overset{}{\sum p_i{x_i}^2}}-{\left(\mathrm{Mean}\right)}^2 \\ =5-4 \\ =1 \end{gathered}$$