1. Consider the function:
fX(x) = {
cx^2, 0 <= x <= 1
c(2 - x)^2, 1 < x <= 2
0, otherwise
(a) Plot/draw fX, then find c to make it the PDF of a random variable X.
(b) What is the mean (expected value) of X?
(c) What is the variance of X?
(d) A median of X is a value m that splits the probability in half:
P(X <= m) = P(X > m) = 1/2
In this problem X has one such m. Show that this median is the same as the mean of X. Does it always have to be (i.e. for other R.V.s)? Give a rough explanation why yes or why no.