Question

Let X denote the sum of the numbers obtained when two fair dice are rolled. Find the 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)`
		\[\sum\limits_{i=1}^{n} x_i\text{P}(x_i)\] = `(252)/(36)` = 7	\[\sum\limits_{i=1}^{n} x_i^2\text{P}(x_i)\] = `(1974)/(36)`

E(X²) = \[\sum\limits_{i=1}^{11} x_i^2\text{P}(x_i)\] = `(1974)/(36)`

Var(X) = E(X²) – [E(X)]²

= `(1974)/(36) - (7)^2`

= `(1974)/(36) - 49`

= `(35)/(6)`
= 5.8333