Question

Let a pair of dice be thrown and the random variable X be the sum of the numbers that appear on the two dice. Find the mean or expectation of X and variance of X

Answer 1

The sample space of the experiment consists of 36 elementary events in the form of ordered pairs (x_i, y_i), where x_i = 1, 2, 3, 4, 5, 6 and y_i = 1, 2, 3, 4, 5, 6.

The random variable X, i.e., the sum of the numbers on the two dice takes the values 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 or 12.

X = xi	P(xi)	xiP(xi)	xi2P(xi)
2	`1/36`	`2/36`	`4/36`
3	`2/36`	`6/36`	`18/36`
4	`3/36`	`12/36`	`48/36`
5	`4/36`	`20/36`	`100/36`
6	`5/36`	`30/36`	`180/36`
7	`6/36`	`42/36`	`294/36`
8	`5/36`	`40/36`	`320/36`
9	`4/36`	`36/36`	`324/36`
10	`3/36`	`30/36`	`300/36`
11	`2/36`	`22/36`	`242/36`
12	`1/36`	`12/36`	`144/36`
		`sum_("i" = 1)^"n"x"P"(x_"i") = 252/36` = 7	`sum_("i" = 1)^"n"x_"i"^2"P"(x_"i") = 1974/36`

∴ E(X) = `sum_("i" = 1)^11x_"i""P"(x_"i")` = 7

E(x²) = `sum_("i" = 1)^"n"x_"i"^2"P"(x_"i") = 1974/36`

Var(X) = E(X²) – [E(X)]² = `1974/36 - (7)^2`

= `1974/36 - 49`

= `35/6`

= 5.83