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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{OSuppose that X is a random variable with density and suppose that ?=e(-4X) Determine ?(?)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.If a dealer’s profit, in units of $5000, on a new automobile can be looked upon as a random variable X having the density function Find the variance of X.Prove 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.Let Z be a standard normal random variable. Find formulae for thedensities of |Z|, Z^2, 1/Z, and 1/Z^2Suppose that Y1, . . . , Yn is a random sample from a population whose density function islet 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 random variable with a uniform distribution over the interval [-2 , 4]. Determine the standard deviation of X.Let X be a continuous random variable with PDF 3 x > 1 x4 fx(x) = otherwise Find the mean and variance of x.SEE MORE QUESTIONS
