Let X1, ..., Xn be a random sample from a population with pdf f(x 1/8,0 < x < θ, zero elsewhere. Let Yi < < Y, be the order statistics. Show that Y/Yn and Yn are independent random variables
8), Let X and Y be continuous random variables with joint density function f(x,y)-4xy for 0 < x < y < 1 Otherwise What is the joint density of U and V Y
10. Let Y1,..., Y, be a random sample from a distribution with pdf 0<y< elsewhere f(x) = { $(0 –» a) Find E(Y). b) Find the method of moments estimator for 8. c) Let X be an estimator of 8. Is it an unbiased estimator? Find the mean square error of X. Show work
4. Let X1,..., X, be a random sample from a population with pdf 0 otherwise Let Xo) <...Xn)be the order statistics. Show that Xu/Xu) and X(n) are independent random variables
Let X and Y be independent exponential random variables with pdfs f(x) = λe-λx (x > 0) and f(y) = µe-µy (y > 0) respectively. (i) Let Z = min(X, Y ). Find f(z), E(Z), and Var(Z). (ii) Let W = max(X, Y ). Find f(w) (it is not an exponential pdf). (iii) Find E(W) (there are two methods - one does not require further integration). (iv) Find Cov(Z,W). (v) Find Var(W).
Problem 1. 15 points] Let X be a uniform random variable in the interval [-1,2]. Let Y be an exponential random variable with mean 2. Assunne X and Y are independent. a) Find the joint sample space. b) Find the joint PDF for X and Y. c) Are X and Y uncorrelated? Justify your answer. d) Find the probability P1-1/4 < X < 1/2 1 Y < 21 e) Calculate E[X2Y2]
Let X, Y be random variables with f(x, y) = 1,-y < x < y, 0 < y < 1. Show that Cov(X,Y) = 0. Are X, Y independent?
Suppose that X and Y are jointly continuous random variables with joint probability density function f(x,y) = {12rºy, 1 0, 0<x<a, 0<y<1 otherwise i) Determine the constant a ii) Find P(0<x<0.5, O Y<0.25) HE) Find the marginal PDFs fex) and y) iv) Find the expected value of X and Y. Le. E(X) and E(Y) v) Are X and Y independent? Justify your answer.
2. Let R be the region R = {(X,Y)|X2 + y2 < 2} and let (X,Y) be a pair of random variables that is distributed uniformly on this region. That is fx,y(x, y) is constant in this region and 0 elsewhere. State the sample space and find the probability that the random variable x2 + y2 is less than 1, P[X2 +Y? < 1].
Let Yı, ..., Yn be an independent and identically distribution sample from the distribution function f(y) = 3y?, for 0 Sy <1. (a) Show the sample mean, y converges in probability to some constant, c. Find c. (b) Find a function that converges in probability to log(C).