1. Consider a random sample of size n from a distribution with CDF F(x) = 1 1/x if1 18. In Exercise 1, find the limiting distribution of n In X1:0.

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Answered: 1. Consider a random sample of size n…

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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3. Draw the cumulative distribution functions of the following distributions: (a) A continuous random variable chosen uniformly from the interval [1,6]. (b) A continuous random variable chosen uniformly from the union [1,2] U [3, 4] U [5, 6]. (c) A discrete random variable chosen uniformly from the set {1, 2, 3, 4, 5, 6}.
(a) Let X1 ~ x² (n, 8) and X2 ~ x² (m, 1). mX1 and its parameters. nX2 (i) Find the distribution of X = (ii) If 1 = 0, name the distribution of X given in 3(a)(i).
iid X1,..., Xn Geo(p) where 0 0; B>0]a conjugate family for samples from the Geometric distributions [Geo(p): 0 0 are known constants. C- Derive the Bayes estimator of p. d- Express the estimator in part (c) as a weighted average of the prior mean of p and the MLE of p.
Suppose X is a discrete random variable which only takes on positive integer values. For the cumulative distribution function associated to X the following values are known: F(18)=0.4F(25)=0.44F(33)=0.5F(41)=0.53F(46)=0.56F(52)=kF(57)=0.64F(65)=0.67 Assuming that Pr[25<X≤52]=0.17, determine the value of k.
Let X₁, X2, X3,..., Xn denote a random sample of size n from the population distributed with the following probability density function: I e) f) g) where k> 0 is a constant. f(x; 2) = 2-kxk-1e (k-1)! if x > 0 elsewhere " Justify whether or not Â is a uniform minimum variance unbiased estimator of 1. Check whether or not the MLE of λ is a consistent. Suggest with a Fisher's factorization theorem, the sufficient estimator of 1.
Assume that X is a random variable whose conditional distribution given the variable Y is poisson P (X | Y) = Po (Y). Suppose further that Y has a gamma distribution Y ∼ Gamma (1, 1). (a) Determine the value E (XY). (b) Determine the conditional distribution P (Y | X).
Suppose X is a random variable of uniform distribution between 1 and 7. Find E(X)
For a random variable, its hazard function also referred to as the instantaneous failure rate is defined as the instantaneous risk (conditional probabilty) that an event of interest will happen in a narrow span of time duration. For a discrete random variable X, its hazard function is defined by the formula hX(k) =P(X=k+ 1|X > k) =pX(k)1−FX(k). For a Poisson distribution with λ= 4.2, find hX(k) and use R to plot the hazard function.
2. Suppose that X1, . Xn are iid Geometric random variables with frequency func- ... tion f(x; 0) = 0(1 – 0)", x = 0, 1, 2, ..., 0 E (0, 1). Find the ML estimator 0, of 0. Show that Ôn is consistent an find its asymptotic distribution.
Let {N(t); t ≥ 0} be a renewal process with the gamma distributed inter- occurrence times with parameters 2 and 3 .The probability distribution of waiting time until the fifteenth renewal occurs is Select one: a normal distribution with paramters 10 and variance 3.33 O b. normal distribution with mean10 and variance 50 gamma distribution with parameters 30 and 3 O d. gamma distribution with parameters 30 and 45 O C.
Consider a random sample of size one (n 1) from the distribution with probability function Soy(-1), if 00 f(y; 0)= 10. otherwise. (a) To test Ho: 0 = 0o versus H₁ : 0 = 0 > 00, find a most powerful test. Let the positive constant to define the rejection region of the test be k. (d) Graph the power function against for k= 0.3, 0.5, 0.8, and comment on how the power of the test changes as the values of k and change. (e) Is the above test in part (a) a uniformly most powerful test? Justify your answer.
2. Consider a random sample X₁, X2,..., Xn of size n from the Uniform (0, 0) distri- bution, where @ > 0. (i) Show that the p.d.f. q(t; 0) of the sufficient statistic is given by T(X)= max X₁ 1

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1. Consider a random sample of size n from a distribution with CDF F(x) = 1 − 1/x if 1