Question

Let X denotes the sum of the numbers obtained when two fair dice are rolled. Find the variance and standard deviation of X.

Answer 1

When two fair dice are rolled, 6 × 6 = 36 observations are obtained.

P(X = 2) = P(1, 1) =1/36

P(X = 3) = P (1, 2) + P(2, 1) =`2/36=1/18`

P(X = 4) = P(1, 3) + P(2, 2) + P(3, 1) =`3/36 = 1/12`

P(X = 5) = P(1, 4) + P(2, 3) + P(3, 2) + P(4, 1) =`4/36 = 1/9`

P(X = 6) = P(1, 5) + P (2, 4) + P(3, 3) + P(4, 2) + P(5, 1) =5/36

P(X = 7) = P(1, 6) + P(2, 5) + P(3, 4) + P(4, 3) + P(5, 2) + P(6, 1)=`6/36 = 1/6`

P(X = 8) = P(2, 6) + P(3, 5) + P(4, 4) + P(5, 3) + P(6, 2) =5/36

P(X = 9) = P(3, 6) + P(4, 5) + P(5, 4) + P(6, 3) =`4/36 =1/9`

P(X = 10) = P(4, 6) + P(5, 5) + P(6, 4) =`3/36 = 1/12`

P(X = 11) = P(5, 6) + P(6, 5) = `2/36 = 1/18`

P(X = 12) = P(6, 6) =1/36

Therefore, the required probability distribution is as follows.

X	2	3	4	5	6	7	8	9	10	11	12
P(X)	1/36	1/18	1/12	1/9	5/36	1/6	5/36	1/9	1/12	1/18	1/36