QUALIFYING EXAMINATION
JANUARY 10, 1994
MATH 519
1. (20 points) Let
X
1
,
X
2
,
X
3
and
X
4
be independent random variables, each uniformly
distributed on the interval (

1
,
1). Find
(a)
P
(
X
1
<X
2
<X
3
<X
4
)
(b)
P
(
X
2
1
<
(
X
1
+
X
2
)
2
)
(c)
P
(
X
2
1
>X
2
2
+
X
2
3
)
2. (20 points) A fair die is rolled repeatedly until it comes up ace. This procedure is
repeated 100 times. Find
(a) the probability that exactly four threes are rolled in exactly 5 of the 100 repe
titions;
(b) the mean and variance of the total number of threes rolled.
3. (20 points) The number of cars arriving at the McDonald’s driveup window in a given
day is a Poisson random variable,
N
, with parameter
λ
. The numbers of passengers
in these cars are independent random variables,
X
i
, each equally likely to be one,
two, three or four. Find the moment generating function of
S
=
N
X
i
=1
X
i
,
the total number of passengers in all the cars.
4. (20 points) Let
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 Spring '09
 Math, Probability theory, total number, X1, Poisson random variable

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