The expectation E(X) of X is 20.
The variance Var(X) of X is 180.
How to find the expectation E(X) of X
(a) To find the expectation E(X) of X, calculate the weighted average of the values of X using their respective probabilities.
E(X) = (0 * P(X=0)) + (10 * P(X=10)) + (20 * P(X=20)) + (30 * P(X=30)) + (40 * P(X=40))
E(X) = (0 * 0.15) + (10 * 0.30) + (20 * 0.10) + (30 * 0.30) + (40 * 0.15)
E(X) = 0 + 3 + 2 + 9 + 6
E(X) = 20
Therefore, the expectation E(X) of X is 20.
(b) To find the variance Var(X) of X, first calculate the squared deviations from the mean (E(X)).
Squared Deviations:

Next, multiply each squared deviation by its respective probability and sum them up:
Var(X) = (400 * 0.15) + (100 * 0.30) + (0 * 0.10) + (100 * 0.30) + (400 * 0.15)
Var(X) = 60 + 30 + 0 + 30 + 60
Var(X) = 180
Therefore, the variance Var(X) of X is 180.