Statistics 134 Fall 2005 Final Exam
Professor James Pitman∗
1. A random variable Xwith values between −1 and 1 has probability density
function f(x) = cx2for xin that range, for some constant c.
(a) Find cas a decimal.
(b) Give a formula for the cumulative distribution function of X.
(c) Find Var(X) as a decimal.
(d) Let Y=X2. 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
Xbe the number of questions the student gets right.
(a) Describe the distribution of Xby 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 Xand Yare independent variables, such that Xhas uniform
distribution on [0,3], and Yhas 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 Xand Y.
4. Dan and Stan each roll a six-sided die. If they roll different numbers, the
one who rolled the higher number wins the difference between the numbers
from the other, in dollars. If they roll the same number, they roll again
until they get different numbers. As soon as they exchange money , that
completes one game. They play 100 games. Let Ribe the number of times
they both have to roll during the i-th game, and let Xibe the amount
that Dan wins in the i-th game. Note that Ri≥1 and that Xican be
negative, but not 0.
(a) Describe the distribution of R1.
∗Transcribed by Chris Pennello on December 21, 2006
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