3. Conditional distributions
Let X be a continuous random variable with pdf f(c). Then, f(a < X < b) is called the conditional distribution of X given a < X < b.
(3a) Show that the cumulative distribution of X conditional on a < X < b is given by
F(x|a < X < b) = (F(x) - F(a))/(F(b) - F(a)) for a < x < b
F(x|a < X < b) = 0 for x < a
F(x|a < X < b) = 1 for x > b
(3b) Differentiate F(x|a < X < b) with respect to x to find the conditional distribution f(a < X < b).
(3c) Show that the conditional expectation of a function of X, say u(X), is given by
E[u(X)|a < X < b] = ∫[a,b] u(x) f(x|a < X < b) dx / ∫[a,b] f(x|a < X < b) dx