Consider the transformation Y= 1/X, and the density function f(x) = 3x² for 01
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Q: Question 1 Consider the transformation Y= 1/X, and the density function f(x) = 3x2 for 0<x<1 and is…
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- Let Y be uniformly distributed in [0, 1], and X = In(Y+1). Find the cumulative distributionfunction of X and the density function of X.Prove that Y₁ and Y₂ are independent given the joint density function - {12192, 4y1y2, 0≤y₁ ≤ 1,0 ≤ y ≤ 1 elsewhere. 0, f(y1, y2) =function f (x, y) =15e-2x-3y a joint probability density function over the range 0c) Let X [0,1]x[0,1]x[0,1]. Let Y = g(X), be a two dimensional random vector, where 91(X ) = X1 + X2 , g2(X) = X2 – X3 . Find the density function of Y. [X1,X2, X3] be a three dimensional random vector uniformly distributed on(11) Let X, Y be independent random variables with X ~ Exp(A1) and Y - Erp(A2). Find the joint density of Z = X +Y,W = X – Y. The following answers are provided. Read the answers carefully before making your choice. (a) The inverse transformation is I = v (z, w) = (z + w)/2, y = v2(z, w) = (z – w)/2. The jacobian of the transformation is the following determinant J = = The bivariate density of Z, W is g(z, w) = A1A2e 1(=+w)/2e¬A2(z¬w)/2 |J], if 쁘 > 0, > 0. This is equivalent to saying that if – z 0, " > 0. This is equivalent to saying that if – z 0, 를 >0. This is equivalent to saying that if - 00 0," > 0. This is equivalent to saying that if – z 0, > 0. This is equivalent to saying that if – z< w < z, 0 < z g(z, w) = if otherwise, (a) (b) (c) (d) (e) N/A (vii- Select One)M 24 Given the joint density function of X, and X, find joint density function of Y, and Y, using linear transformation.2. Consider the joint density function 16M f(x,y) = {#, z> 2,0Let X and Y have the joint probability density function fxy(x, y) = {#², for 0Let X and Y be continuous random variables with a joint probability density function (pdf) of the form f(x, y) = {k(x + y), 0sxsys1 0, elsewhere Find: Show that the value of k = 2 so that f(x, y) is a joint pdf. he marginal of X and Y. the joint cumulative density function (CDF), F(x,y). the conditional pdf of Y given X. E(Y|X = -1)Recommended textbooks for youA First Course in Probability (10th Edition)ProbabilityISBN:9780134753119Author:Sheldon RossPublisher:PEARSONA First Course in Probability (10th Edition)ProbabilityISBN:9780134753119Author:Sheldon RossPublisher:PEARSON