STATISTICS 134
Practice Final
There are 9 questions, worth a total of 49 points. Calculations should be
worked through to an explicit numerical answer. Show your work!
1. [5 points]
Let
U
be a continuous r.v. with uniform distribution on (0
,
1).
Let
X
= log
U
1

U
. Find a formula for the density function
f
(
x
) of
X
.
2. [5 points]
A box contains
n
tags numbered 1
,
2
,...,n
. Two tags are
drawn without replacement, giving two numbers: write
X
for the smaller
and
Y
for the larger number. Calculate
P
(
Y
=
X
+ 1).
3. [5 points]
Consider Poisson random scatter with intensity
λ
on the plane.
Let (
X,Y
) be the coordinates of the random point of the scatter which is
closest to the origin. Find the joint density function
f
(
x,y
) of (
X,Y
).
4. [5 points]
A roulette wheel has 38 slots, of which 18 are red and 18 are
black. In 100 spins of the wheel, let
R
be the number of “reds” and let
B
be
the number of “blacks”. Calculate the correlation cor(
R,B
).
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 Spring '05
 JamesPitman
 Statistics, Probability, Probability theory, Distribution function, probability density function

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