Statistics 134 Fall 2005 Final Exam
Professor James Pitman
*
1. A random variable
X
with values between

1 and 1 has probability density
function
f
(
x
) =
cx
2
for
x
in that range, for some constant
c
.
(a) Find
c
as a decimal.
(b) Give a formula for the cumulative distribution function of
X
.
(c) Find Var(
X
) as a decimal.
(d) Let
Y
=
X
2
. Find the probability density function of
Y
.
2. A multiple choice test has 4 possible answers for each question, exactly
one of which is right. The test has 20 questions. A student knows the
correct answer to 14 questions and guesses at random for the other 6. Let
X
be the number of questions the student gets right.
(a) Describe the distribution of
X
by a formula.
(b) Give a numerical expression for
P
(
X
≥
19).
(c) Evaluate
E
(
X
) as a decimal.
(d) Evaluate Var(
X
) as a decimal.
3. Suppose
X
and
Y
are independent variables, such that
X
has uniform
distribution on [0
,
3], and
Y
has exponential distribution with rate
λ
= 1.
(a) Find
P
(
X < Y
).
(b) Find the probability density function for
Z
:= min(
X,Y
), the mini
mum of
X
and
Y
.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '03
 aldous
 Statistics, Normal Distribution, Probability, Probability theory, probability density function, probability density

Click to edit the document details