Let X₁, X₂, X3, X₁, denote a random sample of size n from the population distributed with thefollowing probability density function:a)b)c)d)e)f)g)f(x; 8) = {0,+((0+1)x, if 0

It is given that f( x; θ ) = { ( θ + 1 )•xθ, 0 < x < 1 0, Elsewhere…

Answered: a) b) c) f(x; 0) = {0, What is the…

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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What is the likelihood ratio p(x|C₁) p(x|C₂) in the case of Gaussian densities?
B2. (a) Consider X₁,..., Xn to be a random sample from the geometric distribution, with probability mass function: P(X= x) = p(1-p), with x = : 0, 1, 2, 3,..., and p € (0, 1]. (i) Using the MGF (M(t) == (ii) Find the Maximum Likelihood Estimator (MLE) for p. -(1-p)et (b) Suppose X₁,..., X₁ is a random sample from a Beta(01, 1) population, and Y₁,..., Ym is an independent random sample from a Beta(02, 1) population. We want to find the approximate Likelihood Ratio Test for Ho: 01 02 00, versus H₁: 01 02. To this aim: = = (i) Under the alternative hypothesis H₁0₁02, show that the MLE for ₁ and 02 are: 0₁ 02 derive E[X] and Var[X]. n Σlog(x)' Recall, that the PDF of Beta(a, b) is fy (y) 00 = - m Σ log(yi) = ['(a) = (a − 1)! for all positive integers a and I (1) = 1) [(a+b) F(a) (b)-1(1-y)b-1 and (ii) Under the null hypothesis Ho: 0₁ 02 = 0o, show that the MLE for 00 is: 01 = n + m E log(xi) + log(yi)
For least-squares to work well, we need: ) the relationship between x and y to be non-linear. residuals to be Uniformly distributed. ) the residuals to have a mean of zero. the residuals to be correlated with the explanatory variable.
2 If X₁ and X₂ are the means of independent ran- dom samples of sizes n₁ and n₂ from a normal population with the mean u and the variance o2, show that the vari- ance of the unbiased estimator 2 w X₁ + (1-w). X₂ is a minimum when @ = n1 n₁ + n₂ idnu
Z is the present-value random variable for a whole life insurance of b payable at the moment of death of (x). You are given: (i) (ii) 8 = 0.05 (iii) The net single premium for this insurance is equal to Var(Z). Calculate b. x+t = 0.01 t≥0 (A) 1.36 (B) 1.68 (C) 2.00 (D) 2.32 (E) 2.64
(4) Consider n i.i.d. samples of X ~ N(µ,0²). Find the maximum likelihood estimate of o?.
Show that the maximum likelihood estimation for the error variance oin linear regression is given by: (see attached)
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 X11 X12, Xini and X21, X22, X2n2 be two independent random samples of size n₁ and n₂ from two normal populations N(₁, 2) and N(2, 2) respectively. (a) Derive the maximum likelihood estimators (mle's) of all the parameters in the first population (X₁). Using analogy, state the mle's of the parameters of the second population. (b) Find the pooled estimator of the common variance when it is assumed that of = = ². Suggest an unbiased estimator of o². |
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4. Consider a random sample {X¡, i = 1, 2, ...n} of size n from a distribution with pdf f(x; 8) = 0(1+ x)-(1+0) if x > 0, and 0 otherwise. (a) Show that the maximum likelihood estimator (mle) ô of 0 is a function of E-1 In(1 + X¡) (b) Let Y = In (1+X), find pdf of Y and mgf of Y. %3D (c) Using result (b) to show that Z = E, In(1 + X;) has a Gamma distribution. Specify the parameters of the gamma distribution (hint: find mgf of Z) (d) For test Ho: 0=1 vs Hi : 0 > 1, we reject null hypothesis if (X1, ..., Xn) € C, where C= {(X, .., Xn), E-, In(1 + X;) > k}. Give an expression to obtain k for given significance level a and what is the expression of power when 0 =1 (no need to integrate the function).

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a) b) c) f(x; 0) = {0, What is the parameter space of 8? Find the maximum likelihood estimator (MLE) of 0. What is the MLE of g (0) ==? 1+8 elsewhere