Posted: September 16th, 2017

SIEO 4150 – Homework 4 Assignment

1. A company makes boxes of various sizes as follows: The length (L) is a rv with
probability mass function P (L = 2) = 0.3, P (L = 4) = 0.7. The width (W) is
a rv with probability mass function P (W = 3) = 0.3, P (W = 4) = 0.4, P (W =
5) = 0.3. The height (H) is a rv with probability mass function P (H = 2) =
1/3, P (H = 3) = 1/3, P (H = 4) = 1/3. Assume that these random variables are
independent.
(a) Find E(W ) and V ar(W ).
(b) What is the expected value of the volume of a box?
(c) What is the variance of the volume of the box?
2. Consider a probability density function for the vector (X, Y ) given by
f (x, y) =

e?x
0

0 ? y ? x < ?,
otherwise.

(a) Verify (by integration) that indeed f de?nes a probability density function on
R2 .
(b) Compute the marginal probability density function of X, fX (x), 0 ? x < ?,
and of Y , fY (y), 0 ? y < ?.
(c) Compute E(X) and E(Y ).
2
2
(d) Compute E(X 2 ) and E(Y 2 ) and then use them to give the variances ?X , ?Y .
(e) Compute ?X,Y = Cov(X, Y ).
(f) Compute the correlation coe?cient ,
?X,Y
?X,Y =
.
?X ?Y
(g) Suppose that X denotes the length L, and Y denotes the width W of a box, and
that the height H is given by a rv independent of (X, Y ) with a distribution
given by P (H = 1) = 0.5, P (H = 4) = 0.2, P (H = 8) = 0.3. Compute the
expected value of the volume of this box.
3. Consider three rvs X1 , X2 , X3 de?ned as follows:
X1 and X2 are iid (independent and identically distributed) with P (?1) = P (1) =
0.5. X3 is de?ned in terms of the other two as follows:
X3 =

X2
0

if X1 = 1,
if X1 = ?1.

(a) Compute the probability mass function (PMF) of X3 . P (0), P (?1), P (1).
(b) Show that X3 and X1 are uncorrelated: E(X3 X1 ) = E(X3 )E(X1 ).
(c) Show, however, that in fact X3 and X1 are not independent:
P (X3 = 1, X1 = 1) = P (X3 = 1)P (X1 = 1).
1

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