Suppose a random variable X has the following probability de

Suppose a random variable X has the following probability density function:

f X ( x ) =x*e^x 0<x<1; 0 otherwise

Let Y = e^X

Find the c.d.f. and the p.d.f. of Y.

Solution

E(x) = int x.f(x) dx =

int x² dx (x from 0 to 1) + int 2x - x² dx (x from 1 to 2) =

1/3 x³ (x from 0 to 1) + x² - 1/3 x³ (x from 1 to 2) =

1/3 + 4 - 8/3 - (1 - 1/3) = 3 -2 = 1

E(x²) = int x².f(x) dx =

int x³ dx (x from 0 to 1) + int 2x² - x³ dx (x from 1 to 2) =

1/4 x⁴ (x from 0 to 1) + 2/3 x³ - 1/4 x⁴ (x from 1 to 2) =

1/4 + 16/3 - 4 - (2/3 - 1/4) = 7/6

Therefore:

Var(x) = E(x²) - (E(x))² = 7/6 - 1 = 1/6