Introduction to Stochastic Processes, Spring 2011
Homework #1
Due: Thursday, January 13th, 2011 in class
1. Recall that a geometric random variable is a discrete one taking values in the set
{
1
,
2
,
3
, . . . ,
}
, with
P
(
X
=
k
) =
p
(1

p
)
k

1
where 0
≤
p
≤
1.
(i) The random number
X
should always be thought of as the number of independent
trials it takes before an experiment succeeds, where
p
is the probability of success
on any given trial. Give a brief explanation of why this is from the formula for
P
(
X
=
k
).
(ii) Compute the mean of
X
.
(iii) Compute the variance of
X
.
2. A bowl contains twenty cherries, exactly fifteen of which have had their stones removed.
A greedy pig eats five whole cherries, picked at random, without remarking on the
presence or absence of stones.
Subsequently, a cherry is picked at random from the
fifteen.
(i) What is the probability that the cherry contains a stone?
(ii) Given that this cherry contains a stone, what is the probability that the pig
consumed at least one stone?
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 Spring '11
 nomane
 Probability theory, Distribution function, lim F, one stone, 365m

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