1. Show that, if X has a U(0,1) distribution, its mean is 1/2 and its variance is 1/12.
Let Y = a + (b - a)X where a, b are constants, a < b. What is the distribution of Y? What are its mean and variance?
2. E(X c) = f x"dx = 1/(k + 1). Hence E(X) = 1/2, E(X2) = 1/3, so Var(X) = 1/12.
Plainly a < Y < b, so let a < y < b; then P(Y < y) = P(a + (b - a)X < y) = P(X < (y-a)/(b-a)) = (y-;)/(b--a), so Y has density 1/(b-a) over (a, b).
Thus Y is FJ-(a,b), E(Y) = a + (b - a)/2 = (a + b)/2 and Var(Y) = (b - a)2/12.