Question

In: Statistics and Probability

Let X be a continuous random variable with a probability density function fX (x) = 2xI...

Let X be a continuous random variable with a probability density function fX (x) = 2xI (0,1) (x) and let it be the function´ Y (x) = e −x

a. Find the expression for the probability density function fY (y).

b. Find the domain of the probability density function fY (y).

Solutions

Expert Solution

Here we can use the variable transformation technique to solve the question.

(a)Here we can use the variable transformation method. Given that the random variable X follows the distribution with pdf,

or .

We want to find out the distribution of .

Since is a monotonically decreasing in x hence we can use the variable transformation technique. The pdf of Y is

Since

Therefore

Therefore

, ( The domain is obtained from part (b))

(b) The domain of Y can be compute directly from domain of X. The domain of X is 0<x<1. The transformation is . The domain of Y is,

and

Hence the domain of Y is

orchestra answered 2 years ago
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