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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