A First Course In Probability, Global Edition
10th Edition
ISBN: 9781292269207
Author: Ross, Sheldon
Publisher: PEARSON
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Chapter 8, Problem 8.10STPE
A tobacco company claims that the amount of nicotine in one of its cigarettes is a random variable with
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-xx0.
B2 If Xfx(x) find the MGF in the case that
fx(x) =
-
1
28
exp{-|x − a\/ẞ},
Use the MGF to compute E(X) and Var(X).
B3 Consider X ~ Bern(p)
(a) Find Mx(t), the moment generating function of X.
iid
(b) If X1,..., Xn
Bern(p), find the MGF, say My (t) of
n
Y =
ΣΧ
(c) Using the fact that
i=1
n
lim (1
(1+2)"=
N→X
= e²
find limn→∞ My (t) in the case that p satisfies limn→∞ np = λ, say.
(d) State the distribution of Y in the case that n is not large, and the distribution of Y in
the limiting case described in the question.
B1 The density of the x2 distribution is given in the notes as
1
F(§)2/2
(x)=()2/21
x/2-1/2, if x > 0, and
e
where I(t)=√xt-¹e dx is the gamma function.
otherwise,
Find the point at which o(a) has its maximum, i.e. find arg max, o, (x)
Chapter 8 Solutions
A First Course In Probability, Global Edition
Ch. 8 - Suppose that X is a random variable with mean and...Ch. 8 - From past experience, a professor knows that the...Ch. 8 - Use the central limit theorem to solve part (c) of...Ch. 8 - Let X1,...,X20 be independent Poisson random...Ch. 8 - Fifty numbers are rounded off to the nearest...Ch. 8 - A die is continually rolled until the total sum of...Ch. 8 - A person has 100 light bulbs whose lifetimes are...Ch. 8 - In Problem 8.7, suppose that it takes a random...Ch. 8 - If X is a gamma random variable with parameters...Ch. 8 - Civil engineers believe that W, the amount of...
Ch. 8 - Many people believe that the daily change of price...Ch. 8 - We have 100 components that we will put in use in...Ch. 8 - Student scores on exams given by a certain...Ch. 8 - A certain component is critical to the operation...Ch. 8 - An insurance company has 10.000 automobile...Ch. 8 - A.J. has 20 jobs that she must do in sequence,...Ch. 8 - Redo Example 5b under the assumption that the...Ch. 8 - Repeat part (a) of Problem 8.2 when it is known...Ch. 8 - A lake contains 4 distinct types of fish. Suppose...Ch. 8 - If X is a nonne9ative random variable with mean...Ch. 8 - Let X be a nonnegative random variable. Prove that...Ch. 8 - Prob. 8.22PCh. 8 - Let X be a Poisson random variable with mean 20....Ch. 8 - Prob. 8.24PCh. 8 - Prob. 8.25PCh. 8 - If f(x) is an Increasing and g(x) is a decreasing...Ch. 8 - If L(p) is the Lorenz curve associated with the...Ch. 8 - Suppose that L(p) is the Lorenz curve associated...Ch. 8 - If X has variance 2, then , the positive square...Ch. 8 - If X has mean and standard deviation , the ratio...Ch. 8 - Compute the measurement signal-to-noise ratio-that...Ch. 8 - Let Zn,n1, be a sequence of random variables and...Ch. 8 - Prob. 8.5TECh. 8 - Prob. 8.6TECh. 8 - Prob. 8.7TECh. 8 - Explain why a gamma random variable with...Ch. 8 - Prob. 8.9TECh. 8 - If X is a Poisson random variable with mean , show...Ch. 8 - Prob. 8.11TECh. 8 - Prob. 8.12TECh. 8 - Prob. 8.13TECh. 8 - Prob. 8.14TECh. 8 - If f and g are density functions that are positive...Ch. 8 - Prob. 8.16TECh. 8 - The number of automobiles sold weekly at a certain...Ch. 8 - Prob. 8.2STPECh. 8 - If E[X]=75E[Y]=75Var(X)=10var(Y)=12cov(X,Y)=3 give...Ch. 8 - Prob. 8.4STPECh. 8 - Prob. 8.5STPECh. 8 - Prob. 8.6STPECh. 8 - Prob. 8.7STPECh. 8 - Prob. 8.8STPECh. 8 - Prob. 8.9STPECh. 8 - A tobacco company claims that the amount of...Ch. 8 - Prob. 8.11STPECh. 8 - Prob. 8.12STPECh. 8 - The strong law of large numbers states that with...Ch. 8 - Each new book donated to a library must be...Ch. 8 - Prove Chebyshevs sum inequality, which says that...
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