The following function is a cumulative distribution function

The following function is a cumulative distribution function (CDF) of mixed type:
F(x) = 0, if x < 0,
= 0.2 + ((x^2)/10), if 0 <= x < 2,
=1, if x >= 2.
(a) Find the constant a and graph the discrete and continuous CDFs F_d(x) and F_c(x) such that F(x) = aF_d(x) + (1-a)F_c(x).
(b) If X_d is a discrete random variable with CDF F_d(x), calculate E(X_d).
(c) If X_c is a continuous random variable with CDF F_c(x), calculate E(X_c).
(d) What is the value of E(X), where X has CDF F(x) given above? Explain your answer.

Solution