STAT/MATH 395 A - PROBABILITY II – UW
Spring Quarter 2016
Néhémy Lim
HW6 : Bivariate Distributions (Part 3) / Distributions of
Functions of Random Variables
Directions. Show and explain all work to receive full credit. Homework is due on Friday, May 20th at the beginning of class.
Problem 1. Consider the set of points G in the two-dimensional plane defined
as :
G = {(x, y) ∈ Z2 | |x| + |y| ≤ 2}
Remember that Z is the set of all integers.
(a) Give the list of all points in G.
(b) Assume that we pick a point from G uniformly at random. Give pXY the
joint probability mass function of X and Y .
(c) Give pX and pY the respective marginal probability mass functions of X
and Y .
(d) Give pX|Y (x|1) the conditional probability mass function of X given Y = 1.
(e) Are X and Y independent?
(f) Find E[X|Y = 1] the expected value of X given Y = 1.
Problem 2. The random variables X and Y have a joint probability density
function given by :
fXY (x, y) = 2e−b(x+y) 1S (x, y)
with S = {(x, y) | 0 < x < y}.
(a) Show that the only value of b that makes fXY a valid joint probability
density function is b = 1.
(b) Compute fX and fY the respective marginal probability density functions
of X and Y .
(c) Are X and Y independent?
(d) If x > 0, compute fY |X (y|x) the conditional probability density function of
Y given X = x.
(e) If x > 0, compute E[Y |X = x] the expected value of Y given X = x.
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Problem 3. Let X with probability density function fX given by:
fX (x) =
and let
x
1[0,2) (x)
2
√
4 − X2
.
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Find fY (y) the probability density function of Y . The answer must include the
support of fY .
Y =1−
Problem 4. Let X be a standard normal random variable N (0, 1) and let
Y = X 2 . Find fY (y) the probability density function of Y . The answer must
include the support of fY .
Problem 5. Let X and Y with joint probability density function fXY given
by:
fXY (x, y) = 1[0,1]2 (x, y)
and let W = X/Y and Z = X +Y . Find fW Z (w, z) the joint probability density
function of W and Z. The answer must include the support of fW Z .
Problem 6. Let X and Y with joint probability density function fXY given
by:
xy
x+y
fXY (x, y) =
exp −
1(0,∞)2 (x, y)
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2
and let W = X/(X + Y ) and Z = X + Y . Find fW Z (w, z) the joint probability
density function of W and Z. The answer must include the support of fW Z .
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