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**Unformatted text preview: **Practice Problems: Computing the density of X + Y .
1) Let X ∼ Exp(λ) and Y ∼ Exp(µ), and let X and Y be independent. Compute the p.d.f.
of X + Y . 1 2) Let X and Y be independent with marginal density functions
2x 0 ≤ x ≤ 1
3/4(1 − y 2 ) −1 ≤ y ≤ 1
fX (x) =
and fY (y ) =
.
0
otherwise
0
otherwise
Compute the density of X + Y . 2 3) Let X and Y be independent random variables such that X ∼ Exp(2) and Y ∼ U (−1, 1).
Compute the p.d.f. of X + Y . 3 4) Let (X, Y ) have joint p.d.f.
x+y
f (x, y ) =
0
a) Are X and Y independent?
b) Compute the p.d.f. of Z = X + Y . 4 0 ≤ x, y ≤ 1
.
otherwise 5) Let X and Y be independent random variables with X ∼ Exp(λ) and Y ∼ Exp(λ).
Compute the p.d.f. of X − Y .
There are two ways to approach this problem. One way is to view Z = −Y as a random
variable and then use convolutions to compute the density of X + Z . The second way to do
this problem is to use the “general method.” That is, calculate P (X − Y ≤ t) and then take
the derivative with respect to t. V-U 5 ...

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