EECS 55 – Engineering Probability
Homework #8
Due: Mar 12, in class
1) If X∼ N (3,4), Y∼ N (−2,6), Z∼ N (0,1), and X,Y, and Zare independent, what
are the distribution, expectation, and variance of S= 3X−2Y+Z?
Notation: N(3,4) is a Normal RV with mean 3 and variance 4.
2) If Xis uniformly distributed over (0,1) and Yis exponentially distributed with
parameter λ= 1 and Xand Yare independent, find the pdf of Z=X+Y.
3) The joint density function of Xand Yis given by f(x, y) = xe−x(y+1),x > 0, y > 0.
a. By just looking at f(x, y), say if Xand Yare independent or not. Explain.
b. Find the conditional density of X, given Y=y. In other words, fX|Y(x|y).
c. Find the conditional density of Y, given X=x.
4) The joint density function of Xand Yis
fX,Y (x, y) = 1
x2y2, x ≥1, y ≥1.
a. Compute the joint density function of U=XY ,V=X/Y .
b. What are the marginal densities of Uand V?
5) Let Xand Ybe random variables with
fX,Y (x, y) = (2e−x−y0≤y≤x < ∞
0 otherwise.
Find E[XY ], Cov(X, Y ), and ρX,Y .
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