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- Let X1, X2, ..., X„ be a random sample from a Uniform(0,0 +1) population with 0 > 0, *(4)"^( +1) be the median of the random sample. X where n is an even integer. Let M Find the probability density function of the median M. Find the expected value and variance of M.Let X and Y be independent Exp(2) random variables. Define W = X + Y. a) Determine the correlation between X and W. The joint pdf of X and W is: fx,w(x, w) = 1^(2) e^(-1w) I{OProve that for the binomial population with density function f (x, p) ="C, p`q" - ', x = 1 , 2,...n. 21. the maximum likelihood estiator for p is . Find its variance.Suppose that two continuous random variables X and Y have joint probability density function fxy = 1sxs2,0sy<3 elsewhere Find the strength of the relationship and interpret the findings.let X and Y be independent standard normal random variables. Determine the pdf of W = X^2 + Y^2. Find the mean and the variance of U= (1/2)WActuary Tong has to study for two actuarial exams: Exam P and Exam FM. The amount of study time that Actuary Tong will spend on each exam in a day follows a continuous random variable that ranges from o to i hour. The amount of study time that Actuary Tong spends on both exams in a day has a joint density function that is equal to the sum of the study times that Actuary Tong spends on ench exam in a day. Caleulate the probability Actuary Tong spends at least half an hour in a day studying for exactly one of the exams. A 0.45 B 0.50 0.55 D 0.60 0.65 REPLet X be a continuous random variable with PDF 3 x > 1 x4 fx(x) = otherwise Find the mean and variance of x.X and Y be standard normal random variables with correlation ρ. Compute the joint and marginal distributions of X + Y and X-Y . Are X + Y and X-Y independent?Suppose X is a random variable taking values in the interval [0,2] with probability density function f(x) = 1-x/2. What is the variance of X?Let Y₁, Y₂,. density function " Yn denote a random sample from a uniform distribution with 1, 0 ≤ y ≤0, f(y|0) = 0, elsewhere. For testing Ho: 000 vs. Ha: 000, show that the a-level Likelihood Ratio Test = (LRT) gives RR = {Y(n) > 00}µ {Y(n) < 0α = }.Let X and Y be two continuous random variables having joint pdffX,Y (x, y) = (1 + XY)/4, −1 ≤x ≤1, −1 ≤y ≤1.Show that X ^2 and Y ^2 are independent.Let X,, X2,... ,X, be a random sample from a distribution with density function if x20 f(x; 0) elsewhere What is the maximum likelihood estimator of 0 ?SEE MORE QUESTIONS
