(a) Since f(x) is a probability distribution, the probabilities given to each value of the random variable in the table must sum to 1. Therefore, we have 0.2+0+u+0.2+0.1 = 1. Solving this equation yields u = 0.5.
(b) E(X) = (-1)(0.2)+(0)(0)+(1)(0.5)+(2)(0.2)+(3)(0.1) = 1
Var(X) = E(X2) - E(X)2 = (-12)(0.2)+(02)(0)+(12)(0.5)+(22)(0.2)+(32)(0.1) - 12 = 1.4
(c) E(Y) = E(X-2) = E(X) - 2 = 1 - 2 = -1
Var(Y) = Var(X-2) = Var(X) = 1.4
Hope this helps!