Q3. (20 points) Let X, X., X, be independent and identically distributed random variables with probability density funct

[answerhappygod]

Q3 20 Points Let X X X Be Independent And Identically Distributed Random Variables With Probability Density Funct 1 (54.65 KiB) Viewed 22 times

Q3 20 Points Let X X X Be Independent And Identically Distributed Random Variables With Probability Density Funct 2 (66.78 KiB) Viewed 22 times

Q3. (20 points) Let X, X., X, be independent and identically distributed random variables with probability density function f(x: 0) = 0x®-1 for x € (0,1) and 8 € (0,0). Note: just for clarity, the pdf could also be written: f(x; 0) = 0x (0-1) I E{X/) = * {+ 1) and V(X) = %@+2)-le + 1].. find the method of moments estimator of Q4. (20 points) Provide your best statistical/mathematical justification for each lettered step in the proof of the Central Limit Theorem given below. Here, let Xx X2 X be independent and identically distributed random variables with LEX/7,2V The Central Limit Theorem states that Z, - , the o/m standard Normal distribution. We will need the following property of (some) moment-generating functions, that you may assume holds (but will have to explain why it works in the proof): MGF Property: My(t) = 1 + E[X]t + E[x4]+/2[1 +0(1)] Proof: The moment generating function of Z, is Mz.) = E[exp{tZnJ] Yn-H) = E exp+ x + X₂ +...+X, пи ſno X = E - no yno ſno X E √no B = (M. MIX:-1/[no] С Xn- E E A A = E(exp{t- = [exp{t "Joc lexp{ **0%) -- exp{*}] E * = ( pronke = ((6) = (1 + 1/n0/2[1 + o(1))" -- exp[/2) = My(t) n C n D Using MGF property, = (1 Which, as n 00, Since the moment generating functions of the Zn converged to the moment generating function of the standard Normal, we proved that the E E

You need not justify elementary algebra or simple calculus steps (e.g., 3x2 - x = x(3x - 1) or Se-dt = e*); however, more complex steps involving gathering like terms or variable substitution should be broken into smaller steps and justified. Sorry to say--if I can't follow the logic, then I take some credit away. 1 X-0 Q1. (20 points, 5 points each part) Suppose that X, X.... are distributed independent and identically distributed continuous random variables supported on (have non-zero pdf on) (0,00) with probability density function f. Assume that f is continuous on (0,00) with lim f(x) = C > 0. Let F denote the corresponding cumulative distribution function and define Yn min {X,X2,...,x) So Y, is the first order statistic or minimum after n trials. In what follows, let e be a positive real number. a. Find P(Yr Se) in terms of F, n, and €. b. Use your answer to part a. to show that Y, converges in probability to (a degenerate random variable placing all probability on) 0. c. Find P(nYSE) in terms of F, n, and e. d. Noting that F(€/n) = C€/n[1 + o(1)], use your answer to part c. to show that ny, converges in law to an exponential random variable. Q2. (20 points) Suppose that X1, X2, ..., X, are independent and identically distributed Geometric random variables with unknown parameter 2 € (0,1),X, E {1,2,3,4,...): f(x:0) = (1 - 0)*-1.0 That is, each X; counts the number of Bernoulli trials, up to and including, the first success, where the probability of success on each of the independent trials is e. Find the maximum likelihood estimator of 8, checking that your solution is in fact a maximum. Is there a special case we need to be careful about for the EX ?