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- 5. A continuous random variable X that can assume value between x-2 and x-5 has a density function given by 1+x f(x) = 8 a. Show that P(2The joint density function of two continuous random variables X and Y is fxy(x, y) = (2x+y) if 22. Let X and Y denote independent random variables with respective probability density func- tions fx(x) = 2x, 0If X, and X, are two random variables having joint density function 2. f(x,,x, ) = +8x,x,* ,0Sx, S 2,0s x,sI 8. Find i. P(X, I) P(X, <1/ X, <0.5) P(X,+ X, <1) ii. iii. iv.Suppose that X and Y are random variables with the joint density function cx² + cy, 0 sx< 3,1Q 4.2. Let (X, Y) be a random variable with the following density: {152-2 0 £x,x (x, y) = { 1. Express E(Y|X) in terms of X. 2. Express Var(XY) in terms of Y. 0 < x < y < 1, otherwise.4. Let X and Y be independent, continuous random variables with densities and fx(x) = = fy (y) = = 0 {} if 0 < x < 2 otherwise. y if 0X (f) = A cos on i+ B sin wn t, where on is a real constant. Find the power Let An and B, be the random variables. A random process is defined as density spectrum of X (t), if A, and Bo are uncorrelated random variables with zero mean and same variance.4) The joint density function of the random variables X and Y is given by (&xy f(x, y) = 0SXS1,0 sysx otherwise Find (a) the marginal density of X, (b) the marginal density of Y, (c) the conditional density of X, (d) the conditional density of Y.Let Y1,..., Yn denote a random sample from the density function given by fy (yla, 0) = F(a)0aY"e3 for y >0, where a > 0 is known and r(-) is the gamma function. Find the MLE of 0.6. Let X and Y be continuous random variables with joint density function 24xy if 0 < x < 1,0 < y < 1 – x f (x, y) = 0. otherwise. Calculate E(Y|X = }).Let Y₁, Y₂,..., Yn be a random sample from the inverse Gaussian distribution with probability density function: f(y, μ, 2) = { Where μ and are unknown. a) What are i. 1 -ACT √ λ 2ny3)že elsewhere if y > 0 the likelihood ii. the log likelihood functions of u and λ. b) Find the maximum likelihood estimators of u and 2.SEE MORE QUESTIONSRecommended textbooks for youMATLAB: An Introduction with ApplicationsStatisticsISBN:9781119256830Author:Amos GilatPublisher:John Wiley & Sons Inc
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