Consider the random variable X whose probability density function f(x) is given by
f(x) = {
0 x < 0
6x 0 <= x < 1/3
3 - 3x 1/3 <= x <= 1
0 x > 1
(a) Compute F(x), the cumulative distribution function of X. [Hint: Break the integral up into regions, i.e. ∫ f(x) dx = ∫ f(x) dx + ∫ f(x) dx. F(x) is continuous.]
(b) Calculate F(1/3).
(c) Using your answer from (b), should the median be higher or lower than 1/3? Why?
(d) Calculate the median value of X.
(e) Calculate the expected value of X. [Hint: break up the integral like in part (a)]
(f) Describe the skew (if any) of f(x).
(g) Calculate η(90), the 90th percentile of X.