Question

Let X represent the number that occurs when a 6-sided red die is tossed and Y the number that occurs when a 6-sided green die is tossed. Find the variance of the random variable 6X-Y.

Answer 1

The value of 6X - Y is computed for different combinations of X, Y as:

	X = 1	X = 2	X = 3	X = 4	X = 5	X = 6
Y = 1	5	11	17	23	29	35
Y = 2	4	10	16	22	28	34
Y = 3	3	9	15	21	27	33
Y = 4	2	8	14	20	26	32
Y = 5	1	7	13	19	25	31
Y = 6	0	6	12	18	24	30

The expected value and second moment of 6X - Y are first computed here as:

E(6X - Y) = 6E(X) - E(Y)

E(X) = E(Y) = (1 + 2 + 3 + 4 + 5 + 6) / 6 = 3.5

Therefore, E(6X - Y) = 6E(X) - E(Y) = 6*3.5 - 3.5 = 5*3.5 = 17.5

Now the second moment is computed here as:

E( (6X - Y)²) = (5² + 4² + .... + 31² + 30²) / 36 = 14910/36 = 414.17

Therefore Var(6X - Y) = E( (6X - Y)²) - [E(6X - Y)]² = 414.17 - 17.5² = 107.9167

Therefore 107.9167 is the required variance here.