A First Course In Probability, Global Edition
10th Edition
ISBN: 9781292269207
Author: Ross, Sheldon
Publisher: PEARSON
expand_more
expand_more
format_list_bulleted
Concept explainers
Videos
Textbook Question
Chapter 4, Problem 4.12TE
Consider n independent sequential trials, each of which is successful with
Expert Solution & Answer
Want to see the full answer?
Check out a sample textbook solution
Students have asked these similar questions
Find the LaPla se trnsofrom of
a) chi-square Distribution.
b) Normal Distribution.
C) Gamma Distribution.
prove that Binomial (n, 2) Poisson (2)
*********************
-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.
Chapter 4 Solutions
A First Course In Probability, Global Edition
Ch. 4 - Two balls are chosen randomly from an urn...Ch. 4 - Two fair dice are rolled, Let X equal the product...Ch. 4 - Three dice are rolled. By assuming that each of...Ch. 4 - Five men and 5 women are ranked according to their...Ch. 4 - Let X represent the difference between the number...Ch. 4 - In Problem 4.5 for n=3, if the coin is assumed...Ch. 4 - Suppose that a die is rolled twice. What are the...Ch. 4 - If the die in Problem 4.7 is assumed fair,...Ch. 4 - Repeat Example 1c, when the balls are selected...Ch. 4 - Let X be the winnings of a gambler. Let...
Ch. 4 - The random variable X is said to follow the...Ch. 4 - In the game of Two-Finger Morra, 2 players show 1...Ch. 4 - A salesman has scheduled two appointments to sell...Ch. 4 - Five distinct numbers are randomly distributed to...Ch. 4 - The National Basketball Association (NBA) draft...Ch. 4 - A deck of n cards numbered 1 through n are to be...Ch. 4 - Suppose that the distribution function of X is...Ch. 4 - Four independent flips of a fair coin are made....Ch. 4 - If the distribution function of X is given...Ch. 4 - A gambling book recommends the following winning...Ch. 4 - Four buses carrying 148 students from the same...Ch. 4 - Suppose that two teams play a series of games that...Ch. 4 - You have $1000, and a certain commodity presently...Ch. 4 - A and B play the following game: A writes down...Ch. 4 - Prob. 4.25PCh. 4 - One of the numbers I through 10 is randomly...Ch. 4 - An insurance company writes a policy to the effect...Ch. 4 - A sample of 3 items is selected at random from a...Ch. 4 - There are two possible causes for a breakdown of a...Ch. 4 - A person tosses a fair coin until a tail appears...Ch. 4 - 4.31. Each night different meteorologists give us...Ch. 4 - To determine whether they have a certain disease,...Ch. 4 - A newsboy purchases papers at 10 cents and sells...Ch. 4 - Prob. 4.34PCh. 4 - A box contains 5 red and 5 blue marbles. Two...Ch. 4 - Consider the friendship network described by...Ch. 4 - Consider Problem 4.22 t with i=2. Find the...Ch. 4 - Find Var (X) and Var (Y) for X and as given in...Ch. 4 - If E[X]=1 and var(X)=5, find a. E[(2+X)2]; b....Ch. 4 - A ball is drawn from an urn containing 3 white and...Ch. 4 - On a multiple-choice exam with 3 possible answers...Ch. 4 - A man claims to have extrasensory perception. As a...Ch. 4 - A and B will take the same 10-question...Ch. 4 - A communications channel transmits the digits 0...Ch. 4 - A satellite system consists of n components and...Ch. 4 - A student is getting ready to take an important...Ch. 4 - Suppose that it takes at least 9 votes from a...Ch. 4 - In some military courts, 9 judges are appointed....Ch. 4 - It is known that diskettes produced by a certain...Ch. 4 - When coin 1 is flipped, it lands on heads with...Ch. 4 - Each member of a population of size n is,...Ch. 4 - In a tournament involving players 1,2,3,4, players...Ch. 4 - Suppose that a biased coin that lands on heads...Ch. 4 - The expected number of typographical errors on a...Ch. 4 - The monthly worldwide average number of airplane...Ch. 4 - Approximately 80000 marriages took place in the...Ch. 4 - State your assumptions. Suppose that the average...Ch. 4 - A certain typing agency employs 2 typists. The...Ch. 4 - How many people are needed so that the probability...Ch. 4 - Suppose that the number of accidents occurring on...Ch. 4 - Compare the Poisson approximation with the correct...Ch. 4 - If you buy a lottery ticket in 50 lotteries, in...Ch. 4 - The number of times that a person contracts a cold...Ch. 4 - The probability of being dealt a full house in a...Ch. 4 - Consider n, independent trials, each of which...Ch. 4 - People enter a gambling casino at a rate of 1...Ch. 4 - The suicide rate in a certain state is 1 suicide...Ch. 4 - Each of 500 soldiers in an army company...Ch. 4 - A total of 2n people, consisting of n married...Ch. 4 - Prob. 4.70PCh. 4 - In response to an attack of 10 missiles, 500...Ch. 4 - A fair coin is flipped 10 times. Find the...Ch. 4 - At time 0, a coin that comes up heads with...Ch. 4 - Consider a roulette wheel consisting of 38 numbers...Ch. 4 - Two athletic teams play a series of games; the...Ch. 4 - Suppose in Problem 4.75 that the two teams are...Ch. 4 - An interviewer is given a list of people she can...Ch. 4 - Prob. 4.78PCh. 4 - Solve the Banach match problem (Example 8e) when...Ch. 4 - In the Banach matchbox problem, find the...Ch. 4 - An urn contains 4 white and 4 black balls. We...Ch. 4 - Suppose that a batch of 100 items contains 6 that...Ch. 4 - A game popular in Nevada gambling casinos is Keno,...Ch. 4 - In Example 81 what percentage of i defective lots...Ch. 4 - A purchaser of transistors buys them in lots of...Ch. 4 - There are three highways in the county. The number...Ch. 4 - Suppose that 10 balls are put into 5 boxes, with...Ch. 4 - There are k types of coupons. Independently of the...Ch. 4 - An urn contains 10 red, S black, and 7 green...Ch. 4 - There are N distinct types of coupons, and each...Ch. 4 - If X has distribution function F, what is the...Ch. 4 - If X has distribution function F, what is the...Ch. 4 - The random variable X is said to have the...Ch. 4 - Let N be a nonnegative integer-valued random...Ch. 4 - Let X be such that P{X=1}=p=1P{X=1}. Find c1 such...Ch. 4 - Let X be a random variable having expected value ...Ch. 4 - Find Var (X) if P(X=a)=(1)=p=1P(X=b)Ch. 4 - Show how the derivation of the binomial...Ch. 4 - Let X be a binomial random variable with...Ch. 4 - Let X be the number of successes that result from...Ch. 4 - Consider n independent sequential trials, each of...Ch. 4 - There are n components lined up in a linear...Ch. 4 - Let X be a binomial random variable with...Ch. 4 - A family has n children with probability pn,n1...Ch. 4 - Suppose that n independent tosses of a coin having...Ch. 4 - Let X be a Poisson random variable with parameter...Ch. 4 - Let X be a Poisson random variable with parameter ...Ch. 4 - Prob. 4.19TECh. 4 - Show that X is a Poisson random variable with...Ch. 4 - Consider n coins, each of which independently...Ch. 4 - From a set of n randomly chosen people, let Eij...Ch. 4 - An urn contains 2 n balls, of which 2 are numbered...Ch. 4 - Consider a random collection of n individuals. In...Ch. 4 - Here is another way to obtain a set of recursive...Ch. 4 - Suppose that the number of events that occur in a...Ch. 4 - Prove i=0nii!=1n!exxndx Hint: Use integration by...Ch. 4 - If X is a geometric random variable, show...Ch. 4 - Let X be a negative binomial random variable with...Ch. 4 - For a hyper geometric random variable,...Ch. 4 - Balls numbered I through N are in an urn. Suppose...Ch. 4 - A jar contains m+n chips, numbered 1, 2,. ., n+m....Ch. 4 - Prob. 4.33TECh. 4 - Prob. 4.34TECh. 4 - Prob. 4.35TECh. 4 - An urn initially contains one red and one blue...Ch. 4 - Prob. 4.37TECh. 4 - Prob. 4.1STPECh. 4 - Prob. 4.2STPECh. 4 - A coin that when flipped comes up heads with...Ch. 4 - Prob. 4.4STPECh. 4 - Suppose that P{X=0}=1P{X=1}. If E[X]=3Var(X), find...Ch. 4 - There are 2 coins in a bin. When one of them is...Ch. 4 - Prob. 4.7STPECh. 4 - Prob. 4.8STPECh. 4 - Prob. 4.9STPECh. 4 - An urn contains n balls numbered 1 through n. If...Ch. 4 - Prob. 4.11STPECh. 4 - Prob. 4.12STPECh. 4 - Each of the members of a 7-judge panel...Ch. 4 - Prob. 4.14STPECh. 4 - The number of eggs laid on a tree leaf by an...Ch. 4 - Each of n boys and n girls, independently and...Ch. 4 - A total of 2n people, consisting of n married...Ch. 4 - Prob. 4.18STPECh. 4 - Prob. 4.19STPECh. 4 - Show that if X is a geometric random variable with...Ch. 4 - Suppose that P{X=a}=p,P{X=b}=1p a. Show that Xbab...Ch. 4 - Prob. 4.22STPECh. 4 - Balls are randomly withdrawn, one at a time...Ch. 4 - Ten balls are to be distributed among 5 urns, with...Ch. 4 - For the match problem (Example 5m in Chapter 2),...Ch. 4 - Let be the probability that a geometric random...Ch. 4 - Two teams will play a series of games, with the...Ch. 4 - An urn has n white and m black balls. Balls are...Ch. 4 - Prob. 4.29STPECh. 4 - If X is a binomial random variable with parameters...Ch. 4 - Let X be the ith smallest number in a random...Ch. 4 - Balls are randomly removed from an urn consisting...
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, probability and related others by exploring similar questions and additional content below.Similar questions
- 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)arrow_forward18. Let X be normally distributed with mean μ = 2,500 and stan- dard deviation σ = 800. a. Find x such that P(X ≤ x) = 0.9382. b. Find x such that P(X>x) = 0.025. ة نفـة C. Find x such that P(2500arrow_forward17. Let X be normally distributed with mean μ = 2.5 and standard deviation σ = 2. a. Find P(X> 7.6). b. Find P(7.4≤x≤ 10.6). 21 C. Find x such that P(X>x) = 0.025. d. Find x such that P(X ≤x≤2.5)= 0.4943. and stan-arrow_forward16. Let X be normally distributed with mean μ = 120 and standard deviation σ = 20. a. Find P(X86). b. Find P(80 ≤x≤ 100). ة ن فـ d. Find x such that P(X ≤x) = 0.40. Find x such that P(X>x) = 0.90.arrow_forwardhe following contingency table details the sex and age distribution of the patients currently registered at a family physician's medical practice. If the doctor sees 17 patients per day, use the binomial formula and the information contained in the table to answer the question: SEX AGE Under 20 20-39 40-59 60-79 80 or over TOTAL Male 5.6% 12.8% 18.4% 14.4% 3.6% 54.8% Female 2.8% 9.6% 13.2% 10.4% 9.2% 45.2% TOTAL 8.4% 22.4% 31.6% 24.8% 12.8% 100.0% if the doctor sees 6 male patients in a day, what is the probability that at most half of them are aged under 39?arrow_forwardPam, Rob and Sam get a cake that is one-third chocolate, one-third vanilla, and one-third strawberry as shown below. They wish to fairly divide the cake using the lone chooser method. Pam likes strawberry twice as much as chocolate or vanilla. Rob only likes chocolate. Sam, the chooser, likes vanilla and strawberry twice as much as chocolate. In the first division, Pam cuts the strawberry piece off and lets Rob choose his favorite piece. Based on that, Rob chooses the chocolate and vanilla parts. Note: All cuts made to the cake shown below are vertical.What pieces would Sam choose based on the Pam and Rob's second division of their own pieces?arrow_forwardPlease plot graphs to represent the functionsarrow_forwardEXAMPLE 6.2 In Example 5.4, we considered the random variables Y₁ (the proportional amount of gasoline stocked at the beginning of a week) and Y2 (the proportional amount of gasoline sold during the week). The joint density function of Y₁ and Y2 is given by 3y1, 0 ≤ y2 yı≤ 1, f(y1, y2) = 0, elsewhere. Find the probability density function for U = Y₁ - Y₂, the proportional amount of gasoline remaining at the end of the week. Use the density function of U to find E(U).arrow_forward7.20 a If U has a x² distribution with v df, find E(U) and V (U). b Using the results of Theorem 7.3, find E(S2) and V (S2) when Y₁, Y2,..., Y, is a random sample from a normal distribution with mean μ and variance o².arrow_forwardAccording to a survey conducted by the American Bar Association, 1 in every 410 Americans is a lawyer, but 1 in every 64 residents of Washington, D.C., is a lawyer. a If you select a random sample of 1500 Americans, what is the approximate probability that the sample contains at least one lawyer? b If the sample is selected from among the residents of Washington, D.C., what is the ap- proximate probability that the sample contains more than 30 lawyers? c If you stand on a Washington, D.C., street corner and interview the first 1000 persons who walked by and 30 say that they are lawyers, does this suggest that the density of lawyers passing the corner exceeds the density within the city? Explain.arrow_forward7.50 Shear strength measurements for spot welds have been found to have standard deviation 10 pounds per square inch (psi). If 100 test welds are to be measured, what is the approximate probability that the sample mean will be within 1 psi of the true population mean? 7.51 Refer to Exercise 7.50. If the standard deviation of shear strength measurements for spot welds is 10 psi, how many test welds should be sampled if we want the sample mean to be within 1 psi of the true mean with probability approximately 992arrow_forward8.12 The reading on a voltage meter connected to a test circuit is uniformly distributed over the interval (0, +1), where 0 is the true but unknown voltage of the circuit. Suppose that Y₁, Y2,..., Y, denote a random sample of such readings. a Show that Y is a biased estimator of and compute the bias. b Find a function of Y that is an unbiased estimator of 0. C Find MSE(Y) when Y is used as an estimator of 0.arrow_forwardarrow_back_iosSEE MORE QUESTIONSarrow_forward_ios
Recommended textbooks for you
College AlgebraAlgebraISBN:9781305115545Author:James Stewart, Lothar Redlin, Saleem WatsonPublisher:Cengage Learning
Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage
College Algebra (MindTap Course List)AlgebraISBN:9781305652231Author:R. David Gustafson, Jeff HughesPublisher:Cengage Learning
Holt Mcdougal Larson Pre-algebra: Student Edition...AlgebraISBN:9780547587776Author:HOLT MCDOUGALPublisher:HOLT MCDOUGAL
Elementary Linear Algebra (MindTap Course List)AlgebraISBN:9781305658004Author:Ron LarsonPublisher:Cengage Learning
College Algebra
Algebra
ISBN:9781305115545
Author:James Stewart, Lothar Redlin, Saleem Watson
Publisher:Cengage Learning
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:9781133382119
Author:Swokowski
Publisher:Cengage
College Algebra (MindTap Course List)
Algebra
ISBN:9781305652231
Author:R. David Gustafson, Jeff Hughes
Publisher:Cengage Learning
Holt Mcdougal Larson Pre-algebra: Student Edition...
Algebra
ISBN:9780547587776
Author:HOLT MCDOUGAL
Publisher:HOLT MCDOUGAL
Elementary Linear Algebra (MindTap Course List)
Algebra
ISBN:9781305658004
Author:Ron Larson
Publisher:Cengage Learning
Mod-01 Lec-01 Discrete probability distributions (Part 1); Author: nptelhrd;https://www.youtube.com/watch?v=6x1pL9Yov1k;License: Standard YouTube License, CC-BY
Discrete Probability Distributions; Author: Learn Something;https://www.youtube.com/watch?v=m9U4UelWLFs;License: Standard YouTube License, CC-BY
Probability Distribution Functions (PMF, PDF, CDF); Author: zedstatistics;https://www.youtube.com/watch?v=YXLVjCKVP7U;License: Standard YouTube License, CC-BY
Discrete Distributions: Binomial, Poisson and Hypergeometric | Statistics for Data Science; Author: Dr. Bharatendra Rai;https://www.youtube.com/watch?v=lHhyy4JMigg;License: Standard Youtube License