Exercise 2. Let X1, X2, , Xn be independent erponential random vari-ables with the same parameter ). Using induction, show that the density ofX = X1+ X2+··....+ Xn isfn (x) =(n- 1)1"-le-(17)%3D( Recall this is the Erlang distribution, a special case of the Gamma distri-bution.)

Answered: Exercise 2. Let X1, X2, , Xn be…

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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Q4\Let a discrete random variable X has values 1, 2, and 3, with probabilities respectively. a) What is PDF for X? b) Sketch the PDF Fx(x) of X. c) Find (i) P(x 2)
"" please answer alll
Let X(n,p) denote binomial random variable with parameter n and p. so that P(X(n,p) > ( greater equal) x) = summation P(X=i) ( summation limit i=x to n ) if W=X(n+1,p) and Y=X(n,p) explay why W>(greater equal "st" Y) See attached for the correct format.
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
Let X, Y be positive random variables (not necessarily independent). Assuming X, Y have all finite moments, write and explain briefly the most appropriate of ≤, ≥, =, ? in the blank, where '?' means no relationship holds in general. i) E(X³). √E(X²) E(X4) ii) P(|X+Y> 2)_ -¹/E((X+Y) ¹) iii) E[√x +3]. iv) E[E[Y²|X]]. 16 √E[(X+3)] (EY)²
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?
b) A continuous random variable X has the p.d.f f(x) = {A(2 – x)(2 + x), 0 < x < 2, Find (i) the value of A, (ii)P(X <1)(iii) P(1 < X <2). l0, otherwis ------------
3 Let X be a random variable with probability Law P (X = r) = q, K for r = 1,2,3.. . 00, Find the moment generating function and mean of ihe random variable X. Let K+ q = 1.
do fast
Example 3.17 Let X be a discrete random variable with range Rx = {0, 7, 5, ,"}, such that Px(0) = Px(;) = Px(5) = Px() = Px(x) = . Find Esin(X).

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Exercise 2. Let X1, X2, , Xn be independent erponential random vari- ables with the same parameter ). Using induction, show that the density of X = X1+ X2+·+ Xn is ... fn (x) = (n- 1)!""e (17) ( Recall this is the Erlang distribution, a special case of the Gamma distri- bution.)