3. Let X1, X2.Xn be a random sample of size n from a Poisson distribution with parameter2. Let T = E, Xi .(a) Use moment generation function technique to determine the distribution of T(b) Find the maximum likelihood estimator for f(0) =e-^ in terms of T

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Answered: 3. Let X1, X2,. Xn be a random sample…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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Without using a moment generating function; Prove that the variance of a beta-distributed random variable with parameters α and β is σ2 = αβ/[(α + β)^2 (α + β + 1)]
a) Let X₁, X₂, X3,..., X, be a random sample of size n from population X. Suppose that X~N(0, 1) you and Y = -√n. VR i) Show that the standard score of the sample mean X, is equal to Y. ii) Show that the mean and variance of the random variable Y are 0 and 1, respectively. iii) Show using the moment generating function technique that Y is a standard normal rando... variable. iv) What is the probability that Y² is between 0.02 and 5.02? b) Let X₁, X₂, X, and Y₁, Y₂.... Y be random samples from populations with moment generating functions Mx, (t) = eat+t² and My (t) = (-), respectively. 25 i) Find the sampling distribution of the statistic W = X₁ + 2X₂ X3 + X₁ + X5. ii) What is the value of the sample size n, if PIX(X-X)2> 68.3392] = 0.025? iii) What is the value of the sample size m, if P(|Y-Hy ≥10) <0.04? QUESTION A
Q5 Find the variance for the PDF px(x) = e-«/2, x > 0.
a) Let X₁, X₂, X₁,...,X, be a random sample of size n from population X. Suppose that X-N(0, 1) and Y = -1-8√n. i) Show that the standard score of the sample mean X, is equal to Y. ii) Show that the mean and variance of the random variable Y are 0 and 1, respectively. iii) Show using the moment generating function technique that Y is a standard normal random variable. iv) What is the probability that Y2 is between 0.02 and 5.02?
Let Y₁, Y2, Y3, Y₁, Y5 be a random sample of size 5 from a standard normal population. Find the moment generating function of the statistic: X = 2Y₁² + Y₂² + 3Y3²+ Y₂² +4Y5² Let Y₁, Y₂, Y3, Y4, Y5 and X₁, X2,..., X, be independent and normally distributed random samples from populations with means ₁ = 2 and ₂ = 8 and variances ₁² = 5 and ₂² = k₁ respectively. Suppose that P(X-Y> 10) = 0.02275, find the value of 0₂² = k.
Suppose the random variable, X, follows a geometric distribution with parameter 0 (0 < 0 < 1). Let X₁, X2,..., X be a random sample of size n from the population of X. (a) Write down the likelihood function of the parameter. (b) Show that the log likelihood function of depends on the sample only through Σj=1 Xj. (c) Find the maximum likelihood estimator (mle) of 0. (d) Find the method of moments estimator (mme) of 0.
Let X1,..., X, a random sample with distribution f(x; 0) = (0 + 1)æº, 0 -1. (a) Find the method of moments estimator of 0. (b) Show that the method of moments estimator is consistent. (c) Find the maximum likelihood estimator of 0. (d) takes value 0 = 1.2. Use the parametric bootstrap method to obtain a 95% revised bootstrap percentile confidence interval using the maximum likelihood estimate. Make sure to include a plot of the bootstrapped values and an interpretation of your confidence interval. Given a sample of size n = 500, suppose that the maximum likelihood estimate of 0
In a continuous-time surplus model, the claim severity is distributed as BN(2, 0.4). Determine the Lundberg upper bound for the probability of ultimate ruin if the initial surplus is 2 and the prIn a continuous-time surplus model, the claim severity is distributed as BN(2, 0.4). Determine the Lundberg upper bound for the probability of ultimate ruin if the initial surplus is 2 and the premium loading factor is 0.25.emium loading factor is 0.25.
Let X₁, X₂,...,Xn be a random sample of size, n from a normal distribution with mean, € and variance, 5. a) b) c) Find the maximum likelihood estimator for 0. Is the estimator obtained in (a) unbiased? Show that the estimator obtained in (a) is a minimum variance unbiased estimator for the parameter, 0.
b) Let X₁, X₂, X, and Y₁, Y₂,...,Y be random samples from populations with moment generating 25 functions Mx, (t) = e³t+t² and My(t) = respectively. i) Find the sampling distribution of the statistic W = X₁ + 2X₂-X₂ + X₁ + X5. ii) What is the value of the sample size n, if PIX(X-X)² > 68.3392] = 0.025? iii) What is the value of the sample size m, if P(|-|≥10) < 0.04?

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3. Let X1, X2,. Xn be a random sample of size n from a Poisson distribution with parameter Li31 (a) Use moment generation function technique to determine the distribution of T (b) Find the maximum likelihood estimator for f(0) =e-d in terms of T