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Let $Z$ be a random variable equal to the sum of two independent random variables $X$ and $Y$ : $Z=X+Y$ The probability density functions of $X$ and $Y$ are given by $f_{X}(x)=a1 [u(x)−u(x−10)]f_{Y}(y)=be_{−10y}u(y) $ a) Find $a$ and $b$. b) Find the probability density function of $Z$. c) What is the probability that $Prob(5≤Z≤20)$.

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 = 10

a is 1/10 and b value is 10.