There are 4 cards numbered 1, 3, 5 and 7, one number on one card. Two cards are drawn at random without replacement. Let X denote the sum of the numbers on the two drawn cards. Find the mean and variance of X.

The 4 cards are numbered as 1, 3, 5, 7.
Let S be the sample space and we have 4 × 3 = 12 elements in sample space.
Here X is the sum of numbers on two drawn cards, X can be 4, 6, 8, 10, 12.
P(X=4) = 2/12 = 1/6
P(X=6) = 2/12 = 1/6
P(X=8) = 4/12 = 1/3
P(X=10) = 2/12 = 1/6
P(X=12) = 2/12 = 1/6
∑PiXi = 4 × 1/6 + 6 × 1/6 + 8 × 1/3 + 10 × 1/6 + 12 × 1/6 = 4 + 6 + 16 + 10 + 12 / 6 = 48 / 6 = 8
Therefore the mean is 8
∑PiXi² = 4² × 1/6 + 6² × 1/6 + 8² × 1/3 + 10² × 1/6 + 12² × 1/6 = 16 + 36 + 128 + 100 + 144 / 6 = 424 / 6 = 212/3
The variance is 212/3 - 8² = 212/3 - 64 = (212 - 192)/3 = 20/3