Math 411 - Fall 2012 - Practice for the final Exam.
1. Pick a point Xuniformly on [0,1] and then pick a point Yuniformly on [0, X].
a. Compute the densities fXand fY|X(y|x).
b. Find the joint density of (X, Y ).
c. Compute the densities fYand fX|Y(x|y).
d. Is it true that Xis uniform on [Y, 1]? Justify your answer.
2. Let Xand Ybe continuous random variables with joint density function
(∗)f(x, y) = ce−y,0< x < y < ∞.
a. Find c.
b. Compute the marginal densities.
c. Is it true that X, Y are independent?
d. Compute the conditional densities fX|Y(x|y), fY|X(y|x).
e. Compute the probability P(0 ≤X≤1|Y= 1).
f. Let Z1, Z2two independent exponentials with parameter 1 each. Let X=Z1
and Y=Z1+Z2. Show that (X, Y ) have joint density as in (∗).
3. The joint probability mass function of the random variables X, Y is
p(0,−1) = p(0,0) = p(0,1) = p(1,0) = 1
4, p(1,−1) = p(1,1) = 0.
a. Compute the marginal probability mass functions of Xand Y.
b. Is it true that Xand Yare independent?
c. Compute EXY , EX, EY.
d. Compte the Cov(X, Y ).
4. Let X, Y, Z be independent and uniformly distributed on (0,1).
a. Compute the probability P(X≤Y+Z
2).
b. Compute the expectation of the random variable Z=eXY2Z.
5. Let X1, X2be independent standard normal variables. Let Y1=1
√2(X1+X2),
Y2=1
√2(X1−X2).
a. Find the joint density of Y1, Y2.
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