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	\[\frac{1}{2}\]
2	\[\frac{1}{2}\]

Computation of mean and variance

xi	pi	pixi	pixi2
1	\[\frac{1}{2}\]	\[\frac{1}{2}\]	\[\frac{1}{2}\]
3	\[\frac{1}{2}\]	\[\frac{3}{2}\]	\[\frac{9}{2}\]
		`∑`pixi = 2	`∑`pixi2 = 5

\[\text{ Mean }  = \sum p_i x_i = 2\]
\[\text{ Variance } = \sum p_i {x_i}2^{}_{} - \left( \text{ Mean}  \right)^2 \]
\[ = 5 - 4\]
\[ = 1\]