A First Course in Probability
9th Edition
ISBN: 9780321794772
Author: Sheldon Ross
Publisher: PEARSON
expand_more
expand_more
format_list_bulleted
Videos
Textbook Question
Chapter 8, Problem 8.23P
Let X be a Poisson random variable with
a. Use the Markov inequality to obtain an upper bound on
b. Use the one-sided Chebyshev inequality to obtain an upper bound on p.
c. Use the Chernoff bound to obtain an upper bound on p.
d. Approximate p by making use of the central limit theorem.
e. Determine p by running an appropriate program.
Expert Solution & Answer
Want to see the full answer?
Check out a sample textbook solution
Students have asked these similar questions
Which statements are true?
Select one or more:
a. Markov’s inequality is only useful if I am interested in that X is larger than its expectation.
b. Chebyshev’s inequality gives better bounds than Markov’s inequality.
c. Markov’s inequality is easier to use.
d. One can prove Chebyshev’s inequality using Markov’s inequality with (X−E(X))2.
Which statements are true?
Select one or more:
a. Markov's inequality is only useful if I am interested in that X is larger than its expectation.
b. Chebyshev's inequality gives better bounds than Markov's inequality.
c. Markov's inequality is easier to use.
d. One can prove Chebyshev's inequality using Markov's inequality with (X-E(X))-.
Markov’s inequality states that
Select one:
a. P(|X|≥t)≤E(X)/t for all random variables X and all t≥0.
b. P(|X|≤t)≤E(X)/t for all random variables X and all t≥0.
c. P(X≥t)≤E(X)/t for all non-negative random variables X and all t≥0.
d. P(X≤t)≤E(X)/t for all non-negative random variables X and all t≥0.
Chapter 8 Solutions
A First Course in Probability
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 - 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 - 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...
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
- Explain how you can determine the steady state matrix X of an absorbing Markov chain by inspection.arrow_forwardConsider the Markov chain whose matrix of transition probabilities P is given in Example 7b. Show that the steady state matrix X depends on the initial state matrix X0 by finding X for each X0. X0=[0.250.250.250.25] b X0=[0.250.250.400.10] Example 7 Finding Steady State Matrices of Absorbing Markov Chains Find the steady state matrix X of each absorbing Markov chain with matrix of transition probabilities P. b.P=[0.500.200.210.300.100.400.200.11]arrow_forwardLet X be a Poisson random variable with mean λ = 20. Estimate the probability P(X ≥25) based on: (a) Markov inequality. (b) Chebyshev inequality. (c) Chernoff bound. (d) Central limit theorem.arrow_forward
- Let X1, ..., X20 be independent Poisson random variables with mean one. 1 2 (a) Use the Markov inequality to obtain a bound on P (X1 + X2 + · · · + X20 > 15). (b) Use the central limit theorem to approximate the above probability.arrow_forwardConsider a linear probability model under the Gauss-Markov assumptions withoutassuming homoskedasticity. This is sufficient to identify a causal effect in the model.TrueFalse A problem of the linear probability model is that an OLS estimation of it can generatefitted values that are negative or greater than one.TrueFalse And both.. otherwise don't ansarrow_forwardLet Zm represent the outcome during the nth roll of a fair dice. Define the Markov chain X, to be the maximum outcome obtained so far after the nth roll, i.e., X, = max {Z1, Z2,..., Zn}. What is the transition probability p22 of the Markov chain {Xn}?arrow_forward
- Let x be a Poisson random variables. Find the probabilities for x using the Poisson Formula.arrow_forwardPlease help.arrow_forwardFind the limiting distribution for this Markov chain. Then give an interpretation of what the first entry of the distribution you found tells you based on the definition of a limiting distribution. Your answer should be written for a non-mathematician and should consist of between 1 and 3 complete sentences without mathematical symbols or terminology.arrow_forward
- For the following Markov models: a ito dove b) find the stationary probability distribution on paper, SUre 5A An ion channel can be in either open or closed state. If it is open, then it has probability 0.1 of closing in 1 microsecond; if closed, it has probability 0.3 of opening in 1 microsecond. 5B An individual can be either susceptible or infected, the probability of infection for a susceptible person is 0.05 per day, and the probability an infected person becoming susceptible is 0.12 per day. 5C The genotype of an organism can be either normal (wild type) or mutant. Each generation, a wild type individual has probability 0.03 of having a mutant offspring, and a mutant has probability 0.005 of having a wild type offspring.arrow_forwardGive an example of a discrete random variable X such that Sx contains exactly two points, and a number c > 0 such that the Markov's inequality is tight, i.e. Е(X) P(X > c) C (You do not need to specify the sample space on which X is defined for this question. The PMF is enough.)arrow_forwardq6- The least-squares estimators follow a normal distribution if the sample size is large, even if the error term is not normally distributed. This is the outcome of which? Select one: a. Gauss-Markov theorem b. Central limit theorem c. Least-squares principle d. none of the abovearrow_forward
arrow_back_ios
SEE MORE QUESTIONS
arrow_forward_ios
Recommended textbooks for you
Elementary Linear Algebra (MindTap Course List)AlgebraISBN:9781305658004Author:Ron LarsonPublisher:Cengage Learning
Linear Algebra: A Modern IntroductionAlgebraISBN:9781285463247Author:David PoolePublisher:Cengage Learning
Elementary Linear Algebra (MindTap Course List)
Algebra
ISBN:9781305658004
Author:Ron Larson
Publisher:Cengage Learning
Linear Algebra: A Modern Introduction
Algebra
ISBN:9781285463247
Author:David Poole
Publisher:Cengage Learning
Finite Math: Markov Chain Example - The Gambler's Ruin; Author: Brandon Foltz;https://www.youtube.com/watch?v=afIhgiHVnj0;License: Standard YouTube License, CC-BY
Introduction: MARKOV PROCESS And MARKOV CHAINS // Short Lecture // Linear Algebra; Author: AfterMath;https://www.youtube.com/watch?v=qK-PUTuUSpw;License: Standard Youtube License
Stochastic process and Markov Chain Model | Transition Probability Matrix (TPM); Author: Dr. Harish Garg;https://www.youtube.com/watch?v=sb4jo4P4ZLI;License: Standard YouTube License, CC-BY