ISyE 2027
Probability with Applications
Fall 2006
R. D. Foley
Homework 6
October 12, 2006
due on Thursday
1. What would be a reasonable guess as to the distribution of (a) the number of cartons of eggs purchased
until you purchase a carton that contains at least one broken egg?
(b) whether the next carton
purchased contains three broken eggs or not?
(c) the number of broken eggs in the next carton
purchased? (d) the number of insects caught by a particular free range chicken between noon and 1
p.m. today?
2. Suppose
X
is a random variable with probability mass function Pr
{
X
=
k
}
=
ck
for
k
= 1
,
2
,
3
,
4.
(a) Determine
c
.
(b) Compute E[
X
].
(c) Compute the probability mass function of
Y
=
X
2
.
(d)
Compute E[
Y
] from the p.m.f. of
Y
. (e) Recompute E[
Y
] = E[
X
2
] using the “law of the unconscious
statistician.” (f) Compute E[
X
(
X

1)] by using the answers from parts (b) and (e) and some basic
properties of E[].
(g) Compute the variance of
X
.
(h) Compute the standard deviation of
X
.
(i)
Compute the conditional probability mass function Pr
{
X
=
k

X
≤
2
}
. (j) Compute E[
X

X
≤
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 Spring '08
 Zahrn
 Probability theory, Probability mass function, R. D. Foley

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