All Matches

Solution Library

Expert Answer

Textbooks

Search Textbook questions, tutors and Books

Oops, something went wrong!

Change your search query and then try again

Result Not Found

Books FREE
Tutors
Study Help
Ask a Question

Search
Sign In
Register

study help
business
introduction to probability statistics

Questions and Answers of Introduction To Probability Statistics

Consider the following random experiment: A fair coin is tossed once. Here, the sample space has only two elements S = {H,T}. We define a sequence of random variables X1, X2, X3, ⋯ on this sample
We perform the following random experiment. We put n ≥ 10 blue balls and n red balls in a bag. We pick 10 balls at random (without replacement) from the bag. Let Xn be the number of blue balls. We
Consider the sample space S = [0, 1] with a probability measure that is uniform on this space, i.e.,Define the sequence {Xn, n = 1, 2,⋯} as follows:Also, define the random variable X on this sample
Find two sequences of random variables {Xn, n = 1, 2,⋯} and {Yn, n = 1, 2,⋯} such thatbut Xn + Yn does not converge in distribution to X +Y. d Xn → X, and d Y→ Y, n
Let X1, X2, X3, ⋯ be a sequence of continuous random variable such thatShow that Xn converges in probability to 0. n fxn (ax) = "e-미피. -ll. ·e 2
Consider a sequence {Xn,n = 1, 2, 3,⋯} such thatShow that Xn a.s.→ 0. Xn n n with probability with probability
Let X1, X2, X3, ⋯ be a sequence of continuous random variable such thatShow that Xn converges in probability to 0. fxn (x) nx² 0 x > otherwise
Let Y1, Y2, Y3, ⋯ be a sequence of i.i.d. random variables with mean EYi = μ and finite variance Var(Yi) = σ2. Define the sequence {Xn,n = 2, 3, . . . } as Xn = Y₁Y2 + Y2Y3+ Yn-1Yn +
Let Y1, Y2, Y3, ⋯ be a sequence of positive i.i.d. random variables with 0 i] = γ n,n = 1, 2, 3, . . . } as Xn = (Y1Y2Y3 · · · Yn-1Yn) = 7, ... Р Show that Xne. for n = 1,2,3,..
Let X1, X2, X3, ⋯ be a sequence of random variable such thatwhere λ > 0 is a constant. Define a new sequence Yn asShow that Yn converges in mean square to λ, i.e., Yn m.s.→ λ. Xn Poisson
Let {Xn,n = 1, 2,⋯} and {Yn,n = 1, 2,⋯} be two sequences of random variables, defined on the sample space S. Suppose that we knowProve that Xn +Yn Lr→ X +Y . You may want to use Minkowski's
Let X1, X2, X3, ⋯ be a sequence of random variable such that Xn ∼ Rayleigh(1), a.s. Show that X → 0. - {n²z exp{-} {" 0 fx, (x) = x > 0 otherwise
Let Y1,Y2, ⋯ be independent random variables, where Yn ∼ Bernoulli (n + 1) for n = 1, 2, 3,⋯. We define the sequence {Xn,n = 2, 3, 4,⋯} as Show that X → 0. a.s. → Xn+1=Y₁ Y₂Y3
Let X be the weight of a randomly chosen individual from a population of adult men. In order to estimate the mean and variance of X, we observe a random sample X1,X2,⋯ ,X10. Thus, the Xi's are
Let X1, X2, X3, X4 be a random sample from the Uniform(0, 1) distribution, and let X(1), X(2), X(3) , X(4). Find the PDFs of X(1), X(2), and X(4).
Let X1, X2, X3, . . ., Xn be a random sample with unknown mean EXi = μ, and unknown variance Var(Xi) = σ2. Suppose that we would like to estimate θ = μ2. We define the estimator ^ Θ asto
Let Xi be i.i.d. Uniform(0, 1). We define the sample mean asa. Find E[Mn] and Var(Mn) as a function of n.b. Using Chebyshev's inequality, find an upper bound onc. Using your bound, show that Mn X₁
Let X be a random variable with EX = 1 and RX = (0, 2). If Y = X3 −6X2, show that EY ≤ −5.
Let X be a positive random variable with EX = 10. What can you say about the following quantities?1. E[X −X3]2. E[Xln√X]3. E[|2 −X|]
Prove for two random variables X and Y with finite moments, and 1 ≤ p Note thatThereforeNow, apply Hölder's inequality. |X+YP = |X+Y|P-¹|X+Y| ≤X+YP ¹ (X+Y) ≤X + YP ¹|X| + |X + YP-¹|Y|.
Let X ∼ Binomial(n, p). Using Markov's inequality, find an upper bound on P(X ≥ αn), where p < α < 1. Evaluate the bound for p = 1/2 and α = 3/4.
in Geometric (p). Using Chebyshev's inequality find an upper bound for P(|X −EX| ≥ b).
Let X ∼ Binomial(n, p). Using Chebyshev's inequality, find an upper bound on P(X ≥ αn), where p < α < 1. Evaluate the bound for p = 1/2 and α = 3/4.
Let X be a random variable with EX = 0 and Var(X) = σ2. We would like to prove that for any a > 0, we haveThis inequality is sometimes called the one-sided Chebyshev inequality. One way to show
Let X ∼ Binomial (n, p). Using Chernoff bounds, find an upper bound on P(X ≥ αn), where p < α < 1. Evaluate the bound for p = 1/2 and α = 3/4.
The number of customers visiting a store during a day is a random variable with mean EX = 100 and variance Var(X) = 225.1. Using Chebyshev's inequality, find an upper bound for having more than 120
Using the Cauchy-Schwarz inequality, show that for any two random variables X and Y |ρ(X,Y)| ≤ 1.Also, |ρ(X,Y )| = 1 if and only if Y = aX +b for some constants a, b ∈ R.
Let Xi be i.i.d. and Xi ∼ Exponential(λ). Using Chernoff bounds find an upper bound for P(X1 +X2 +⋯+Xn ≥ a), where a > n/λ. Show that the bound goes to zero exponentially fast as a function
Let X be a positive random variable. Compare E[Xa] with (E[X])a for all values of a ∈ R.
Let X and Y be two independent Uniform (0, 1) random variables. Let the random vectors U and V be defined asDetermine whether CU and CV are positive definite. U= [X²Y] · V= [X + x] X Y LX+Y +
Let X and Y be two jointly continuous random variables with joint PDFand let the random vector U be defined as1. Find the mean vector of U, EU.2. Find the correlation matrix of U, RU.3. Find the
Let X and Y be two jointly continuous random variables with joint PDFand let the random vector U be defined asFind the correlation and covariance matrices of U. fx,y(x, y) = 3 ت ادا 0 x² + y 0 <
Let X be a random variable with characteristic function ϕX(ω). If Y = aX +b, show that ϕY (ω) = ejωbϕX(aω).
Let X1, X2, X3, . . ., Xn be a random sample from a distribution with mean EXi = θ, and variance Var(Xi) = σ2. Consider the following two estimators for θ:Find MSE(^Θ1) and MSE(^Θ2) and show
Let X1,…,X4 be a random sample from a Geometric(p) distribution. Suppose we observed (x1,x2,x3,x4) = (2, 3, 3, 5). Find the likelihood function using PXi(xi; p) = p(1 −p)xi−1 as the PMF.
Let X1, X2, X3, . . ., Xn be a random sample with mean EXi = θ, and variance Var(Xi) = σ2. Show that ^Θn = X̅ is a consistent estimator of θ.
Let X1,…,X4 be a random sample from an Exponential (θ) distribution. Suppose we observed (x1,x2,x3,x4) = (2.35, 1.55, 3.25, 2.65). Find the likelihood function usingas the PDF. fx; (x₁; 0) =
Let X1, X2, X3, . . ., Xn be a random sample with mean EXi = μ, and variance Var(Xi) = σ2. Suppose that we useto estimate σ2. Find the bias of this estimator 1 5--2-xr-|(2x-ux) (X X)
Often when working with maximum likelihood functions, out of ease we maximize the log-likelihood rather than the likelihood to find the maximum likelihood estimator. Why is maximizing L(x; θ) as a
Let T be the time that is needed for a specific task in a factory to be completed. In order to estimate the mean and variance of T, we observe a random sample T1,T2,⋯, T6. Thus, Ti's are i.i.d. and
Let X be one observation from a N(0,σ2) distribution.a. Find an unbiased estimator of σ2.b. Find the log likelihood, log(L(x;σ2)), usingas the PDF.c. Find the Maximum Likelihood Estimate (MLE) for
In this problem, we would like to find the CDFs of the order statistics. Let X1,…,Xn be a random sample from a continuous distribution with CDF FX(x) and PDF fX(x).Define X(1),…,X(n) as the order
For the following random samples, find the likelihood function:1. Xi ∼ Binomial(3, θ), and we have observed (x1,x2,x3,x4) = (1, 3, 2, 2).2. Xi ∼ Exponential(θ) and we have observed
Let X1,…,Xn be a random sample from a Poisson(λ) distribution.a. Find the likelihood equation, L(x1,…,xn;λ), usingas the PMF.b. Find the log likelihood function and use that to obtain the MLE
Suppose that we have observed the random sample X1, X2, X3, . . ., Xn, where Xi ∼ N(θ1, θ2), soFind the maximum likelihood estimators for θ1 and θ2. fx; (xi; 01,02)
A random sample X1, X2, X3, . . ., X100 is given from a distribution with known variance Var(Xi) = 81. For the observed sample, the sample mean is X̅ = 50.1. Find an approximate 95% confidence
For the following examples, find the maximum likelihood estimator (MLE) of θ:1. Xi ∼ Binomial(m, θ), and we have observed X1, X2, X3, . . ., Xn. 2. Xi ∼ Exponential(θ) and we have observed
In this problem, we would like to find the PDFs of order statistics. Let X1,…,Xn be a random sample from a continuous distribution with CDF FX(x) and PDF fX(x). Define X(1),…,X(n) as the order
To estimate the portion of voters who plan to vote for Candidate A in an election, a random sample of size n from the voters is chosen. The sampling is done with replacement. Let θ be the portion of
Let Z ∼ N(0, 1), find xl and xh such that P ( ₁₁ ≤ 2 ≤ 2₂ ) = 0 xh = 0.95
Let X1, X2, X3, . . ., X100 be a random sample from a distribution with unknown variance Var(Xi) = σ2
Let X1, X2, X3, . . ., Xn be a random sample from a normal distribution N(θ, 1). Find a 95% confidence interval for θ.
A random sample X1, X2, X3, . . ., X36 is given from a normal distribution with unknown mean μ = EXi and unknown variance Var(Xi) = σ2. For the observed sample, the sample mean is X̅ = 35.8, and
Check that the random variables Q1 = X̅ −θ and Q2 = √n(X̅ − θ) are both valid pivots in Example 8.13.Example 8.13:Let X1, X2, X3, . . ., Xn be a random sample from a normal distribution
Let X1, X2, X3, X4, X5 be a random sample from a N(μ, 1) distribution, where μ is unknown. Suppose that we have observed the following values 5.45, 4.23, 7.22, 6.94, 5.98We would like to decide
Let X1, X2, X3, . . ., Xn be a random sample from a distribution with known variance Var(Xi) = σ2, and unknown mean EXi = θ. Find a (1 −α) confidence interval for θ. Assume that n is large.
Let X1,…,X9 be a random sample from a N(μ, 1) distribution, where μ is unknown. Suppose that we have observed the following values 16.34, 18.57, 18.22, 16.94, 15.98, 15.23, 17.22, 16.54, 17.54We
An engineer is measuring a quantity θ. It is assumed that there is a random error in each measurement, so the engineer will take n measurements and report the average of the measurements as the
Let X1, X2 ,..., X150 be a random sample from an unknown distribution. After observing this sample, the sample mean and the sample variance are calculated to beDesign a level 0.05 test to choose
We would like to estimate the portion of people who plan to vote for Candidate A in an upcoming election. It is assumed that the number of voters is large, and θ is the portion of voters who plan to
Let X1, X2, X3, X4, X5 be a random sample from a N(μ,σ2) distribution, where μ and σ are both unknown. Suppose that we have observed the following values 27.72, 22.24, 32.86, 19.66, 35.34. We
Let X1, X2, X3, . . ., Xn be a random sample from this distribution, which means that the Xi's are i.i.d. and Xi ∼ Bernoulli(θ). In other words, we randomly select n voters (with replacement) and
Let X1, X2 ,..., X121 be a random sample from an unknown distribution. After observing this sample, the sample mean and the sample variance are calculated to beDesign a test to decide betweenand
There are two candidates in a presidential election: Candidate A and Candidate B. Let θ be the portion of people who plan to vote for Candidate A. Our goal is to find a confidence interval for θ.
Consider the following observed values of (xi, yi):a. Find the estimated regression linebased on the observed data.b. For each xi, compute the fitted value of yi usingc. Compute the residuals, ei =
Suppose we would like to test the hypothesis that at least 10% of students suffer from allergies. We collect a random sample of 225 students and 21 of them suffer from allergies.a. State the null and
A farmer weigh s 10 randomly chosen watermelons from his farm and he obtains the following values (in lbs): 7.72 9.58 12.38 7.77 11.27 8.80 11.10 7.80 10.17 6.00Assuming that the weight is normally
Consider the simple linear regression modelwhere ϵi's are independent N(0,σ2) random variables. Therefore, Yi is a normal random variable with mean β0 +β1xi and variance σ2. Moreover, Yi's are
Consider the following observed values of (xi, yi): (1, 3), (3, 7).a. Find the estimated regression linebased on the observed data.b. For each xi, compute the fitted value of yi usingc. Compute the
For the data given in Example 8.20, find a 95% confidence interval for σ2. Again, assume that the weight is normally distributed with mean μ and and variance σ, where μ and σ are unknown.Example
You have a coin and you would like to check whether it is fair or not. More specifically, let θ be the probability of heads, θ = P(H). You have two hypotheses: . 0 = 0o = 1/². Ho (the null
The average adult male height in a certain country is 170 cm. We suspect that the men in a certain city in that country might have a different average height due to some environmental factors. We
Let X1, X2,...,Xn be a random sample from a N(μ,σ2) distribution, where μ is unknown and σ is known. Design a level α test to choose between Ho: Ηο: μ Σμο, Η: μ > μο.
You have a coin and you would like to check whether it is fair or biased. More specifically, let θ be the probability of heads, θ = P(H). Suppose that you need to choose between the following
Here, we look again at the radar problem (Example 8.23). More specifically, we observe the random variable X: X = θ + W, where W ∼ N(0,σ2 = 1/9). We need to decide betweenLet X = x. Design a
Let X1, X2,...,Xn be a random sample from a N(μ, σ2) distribution, where μ and σ are unknown. Design a level α test to choose betweenH0: μ = μ0,H1: μ ≠ μ0.
For the above example (Example 8.24), find β, the probability of type II error, as a function of μ.Example 8.24 Let X1, X2,...,Xn be a random sample from a N(μ,σ2) distribution, where μ is
Let X1,X2,...,Xn be a random sample from a N(μ,σ2) distribution, where μ is unknown but σ is known. Design a level α test to choose betweenH0: μ = μ0,H1: μ ≠ μ0.
Again consider the simple linear regression modelwhere ϵi's are independent N(0,σ2) random variables, and Y = Bo + Britai
Consider a radar system that uses radio waves to detect aircraft. The system receives a signal and, based on the received signal, it needs to decide whether an aircraft is present or not. Let X be
Consider the following observed values of (xi, yi): (1, 3) (2, 4) (3, 8) (4, 9)1. Find the estimated regression linebased on the observed data.2. For each xi, compute the fitted value of yi using3.
For the data in Example 8.31, find the coefficient of determination.Example 8.31Consider the fo llowing observed values of (xi, yi):(1, 3) (2, 4) (3, 8) (4, 9)

Showing 1400 - 1500 of 1482

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15

Get In Touch

About Us
Contact Us
Career
Jobs
FAQ
Campus Ambassador

Company Info

Security
Copyrights
Privacy Policy
Terms & Condition
SolutionInn Fee
Scholarship

Services

Sitemap
Fun
Definitions
Become Tutor
Study Help Categories
Recent Questions
Expert Questions

Copyright © 2024 SolutionInn All Rights Reserved.

Get the SolutionInn - Study Help App