Let X be a continuous random variable with uniform distribut

Let X be a continuous random variable with uniform distribution on the interval (0, 1). Find the probability density function (PDF) of the random variable Y defined by

Let X be a continuous random variable with uniform distribution on the interval (0, 1). Find the probability density function (PDF) of the random variable Y defined by Y = {1}/{lambda } ln(1-X), lambda > 0

Solution

Let X have probability density function given by
f(x) = 3x2,
with domain [0, 1]. Find E(X).
Solution
We have

E(X) = ab x f(x) dx.
= 01 (x)(3x2) dx
= 01 (3x3) dx
= [3x4/4]01 = 3/4.
Thus, the expected value of X is 3/4.