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(Solved): Let Z be a random variable equal to the sum of two independent random variables X and Y : Z=X+Y Th ...
![Let \( Z \) be a random variable equal to the sum of two independent random variables \( X \) and \( Y \) :
\[
Z=X+Y
\]
The p](https://media.cheggcdn.com/media/b54/b54c9042-a019-496c-9b04-9b12a4eb6628/phphGo2Tz)
Let be a random variable equal to the sum of two independent random variables and : The probability density functions of and are given by a) Find and . b) Find the probability density function of . c) What is the probability that .
Expert Answer
a) To find the values of a and b, we need to ensure that the probability density functions (PDFs) of X and Y integrate to 1 over their respective support intervals.Given: To find a, we integrate over its support interval : so when , and 0 otherwise, we have: To find b, we integrate over its support interval : Since u(y) = 1 when y ? 0, and 0 otherwise, we have: Applying integration, we get: b = 10a is 1/10 and b value is 10.