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 varianc

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	\[\frac{1}{6}\]	\[\frac{12}{6}\]	\[\frac{144}{6}\]
		`∑`pixi = 7	`∑`pixi2=\[\frac{364}{6}\]

\[\text{ Mean }  = \sum p_i x_i = 7\]
\[\text{ Variance } = \sum p_i {x_i}2^{}_{} - \left( \text{ Mean }  \right)^2 \]
\[ = 60 . 7 - 49\]
\[ = 11 . 7\]