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- Let X be a random variable with pdf f(x) = kx*,-1Exp(2), compute the If X is an exponential random variable with parameter A = 2, i.e., X ~ probability density function of the random variable Y = In X.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?2. Let X be a continuous random variable with probability density function (pdf) 3x�� 0(47) Let X z b(8,-) find E(5+6x) and distribution function.Let Y₁, Y2,..., Y₁ denote a random sample from the probability density function f(y) = {0y⁰-1₁ 0 0. Show that Y is a consistent estimator of 0/(0+1).(b) (i) Briefly explain how moment generating function of a random variable, y may be used to Generate its moments. (ii) Let the random variable, y have probability density function S(t) = kye-1/2y y20 , elsewhere Where k> 0 is a constant. Find the moment generating function of y and use it to evaluate the mean and variance of y leaving your results in terms of k.3. Let X and Y be continuous random variables with joint PDF (3x 0s ysxs1 f(x, y) = {* otherwise Determine the correlation of variables X and Y.Let X and Y be two continuous random variables with joint probability density [3x function given by: f(x,y)= 0Recommended textbooks for youCalculus For The Life SciencesCalculusISBN:9780321964038Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.Publisher:Pearson Addison Wesley,Glencoe Algebra 1, Student Edition, 9780079039897…AlgebraISBN:9780079039897Author:CarterPublisher:McGraw HillAlgebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:CengageTrigonometry (MindTap Course List)TrigonometryISBN:9781337278461Author:Ron LarsonPublisher:Cengage LearningCalculus For The Life SciencesCalculusISBN:9780321964038Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.Publisher:Pearson Addison Wesley,Glencoe Algebra 1, Student Edition, 9780079039897…AlgebraISBN:9780079039897Author:CarterPublisher:McGraw HillAlgebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:CengageTrigonometry (MindTap Course List)TrigonometryISBN:9781337278461Author:Ron LarsonPublisher:Cengage Learning