31. If X is a continuous random variable with p.d.f. f(x), the pth percentile of x (sometimes called thepth percentile of the population) is defined as that value xp for which P(X ≤ xp) =

 xp

−∞

f(x)dx = p/100 The 50th percentile is called the population median. Also, suppose that a continuous random variable has cumulative distribution F(x) =

⎧⎪⎨

⎪⎩

0, x>1 xn, 0 < x ≤ 1 1, x>1 where n ≥ 1. Then,

(a) Find the probability density function f(x).

(b) Find the median of X.

(c) Find the mean and variance of X.

224 5 Continuous Random Variables