Problem 5 A₁ = {wen: i-1≤X(w)
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- Problem A₁ = {wen: Let X be random variable which takes non-negative values only. Let i-1≤X(w)Suppose that Z is a discrete random variable with positive expectation. If Var(Z) = 3 and E(Z²) = 4, match up the following quantities. E(1 - 4Z) Drag answer here E(Z² - 5) Drag answer here Var(Z² – 5) Drag answer here Var(2Z) Drag answer here Var(2 - Z) Drag answer here 3 Cannot be determine d 12 -1 -33. The expectation operator E can be applied to a random vector. Specif- ically, if X = [X1 X2 …… Xn]", then EX = [EX1 EX2 .… EXn]". Show that E[(X – EX)(X – EX)"]= [cov(X¡, X;)]"j=1 •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.Question 3. Let X₁,, Xn be a random sample from a distribution with the pdf given by f₁₁x(x) = exp(-ª) if x ≥ A, otherwise f(x) = 0, where > 0. Find the MLE's of 0 and X. Start by writing the likelihood function and note the constraint involving A. Question 4. #4.2.9 of the textbook.Example 2.19 Let X be a random variable with possible values {x1, x2,...x) such that P(X = x) =!v i. Find E(X).1. Random variable X has p.d.f -6z 6e fx(z) otherwise and let Y = eX. Calculate E(X)Part II: Sections 2.1 - 2.7 8. Assume that X is a geometric random variable with p=0.32. (a) Compute P(X > 13[X > 3). (b) Compute E(X²).3. Suppose X is a discrete random variable with pmf defined as p(x) = log10 ( for %3D x = {1,2,3,...9} Prove that p(x) is a legitimate pmf.4. Using the inverse CDF method, find formulae for generating random variables having the following PDFs: (a) f(x) = 2 cos x 3 sin³ x " (b) f(x) = 8x (x + 1)³ 0≤x≤1. 3"Suppose that X1, X2, ..., Xn is a random sample with a probability law P(x) = {0x, 0 Where 8 is an unknown parameter. si x = 1,2,3, 4; otherwise A. Get an estimator of 0 by the method of the moments. B. Prove that the estimator is unbiased.4.Suppose that X is a random variable for which E(X)=µ and Var(X)=o². Show that E(X(X-1)]=µ(µ-1)+o?SEE MORE QUESTIONS