Probability Q2 Thx

(a) 1/2 + k + 1/10 + 3/50 + 1/25 = 1
25/50 + k + 5/50 + 3/50 + 2/50 = 1
so (25 + 5 + 3 + 2)/50 + k = 1
35/50 + k = 1, so k = 15/50 = 3/10.
(b) Expected value of X = 1/2(1) + 3/10(2) + 1/10(3) + 3/50(4) + 1/25(5)
= 0.5 + 0.6 + 0.3 + 0.24 + 0.2 = 1.84
(c) E(X^2) = 1/2(1^2) + 3/10(2^2) + 1/10(3^2) + 3/50(4^2) + 1/25(5^2)
= 0.5 + 1.2 + 0.9 + 0.96 + 1 = 4.56
So Variance, Var(X) = 4.56 - 1.84^2 = 1.1744
(d) Profit of 1 pizza = 125 - 50 = 75
so expected profit = $75 x 1.84 = $138.