# Probability (P1 Exam Parctice) - Comparing Similar Transformation Problems

This section I think can be difficult because it is not always clear what you have to do. Here are two similar problems plus another for general practice.

What I learned is that, in example 1, we were given the CDF, or cumulative distribution function, so all we had to do was take the derivative to get the probability for y = 2. In example 2, we had to multiply the PDF with the derivative because we were not given the CDF and need the Constant from the integral.

## Example 1

If X is an exponential random variable with mean 2, and , find , where denotes the density of Y.

Inverse:

Exponential Function Cumulative:

Derivative of Above Function Gives PDF:

Plug in 2:

## Example 2

The density of X is given by for all real x-values (negative infinity to positive infinity). Find the density of Y = 1 / X.

Inverse:

Density (plug in inverse):

Derivative of Inverse:

Multiply:

## Example 3

The cost of damage C in a fire has a density given by for c > 0. If the damage is insured with a $500 deductible, what is the expected payment?

Since we are finding the expected value after a $500 deductible we multiply the density function by (c-500) and integrate from 500 to infinity.

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