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- If F(x1, x2, x3) is the value of the joint distribution function of X1, X2, and X3 at (x1, x2, x3), show that the joint marginal distribution function of X1 and X3 is given by M(x1, x3) = F(x1, q, x3) for −q < x1 < q, −q < x3 < q and that the marginal distribution function of X1 is given by G(x1) = F(x1, q, q) for −q < x1 < q With reference to Example 19, use these results to find (a) the joint marginal distribution function of X1 and X3; (b) the marginal distribution function of X1.For the joint distribution function f(x,y)=2 for x≥0,y≥0,x+y≤1 and 0 otherwise Find the marginal of X and Y. Are X and Y independent? Find E(X )IF F(x, y) is the value of the joint distribution function of X and Y at (x, y), show that the marginal distribution function of X is given by G(x) = F(x, ∞) for - ∞ <x < ∞ Use this result to find the marginal distribution function of X for the random variable F(x, y) = { (1-e-x2 ) (1- e-y2) for x>0, y> 0 and 0 elsewhere
- Let X and Y have the joint pdff(x,y)=cx^2y,−y≤x≤1.5,0≤y≤1.5 a) Determine the value of the constant cc b) Give the value of the marginal pdf fX(x) at x = -0.5 c) Give the value of the marginal cdf FX(x) at x = 0.5 d) Give the value of the conditional distributionfX|Y(x|Y=0.5) at x = 0.5If X1, X2, ... , Xk have the multinomial distribution of Definition 8, show that the mean of the marginal distribu-tion of Xi is nθi for i = 1, 2, ... , k.Find marginal distributions P(x) and P(y)
- For the joint distribution function f(x,y)= 2/5(x+y2) for 0≤x≤1, 0≤y≤1 and 0 otherwise Find P(X+Y<1)LetX1,X2,...,Xn be a sequence of independent and identically distributed random variables having the Exponential(λ) distribution,λ >0, fXi(x) ={λe−λx, x >0 0, otherwise Define the random variable Y=X1+X2+···+Xn. Find E(Y),Var(Y)and the moment generating function ofY.Let X and Y have the joint pdff(x,y)=cx^2y,−y≤x≤1.5,0≤y≤1.5f(x,y)=cx2y a) Determine the value of the constant cc b) Give the value of the marginal pdf fX(x)at x = -0.5 c) Give the value of the marginal cdf FX(x)at x = 0.5 d) Give the value of the conditional distribution fX|Y(x|Y=0.5) at x = 0.5
- Consider X and Y are joint distributed with PDFf(x,y)=x+y, 0≤x≤1, 0≤y≤1. (d) Find the marginal distributions of X and Y .(e) Find the conditional distribution of X given Y = y.Consider the joint PDF f(x,y)=4xy for 0<x<1 and 0<y<1. (a) Are X and Y independent?(b) Find both marginal distributions. (c) Find both conditional distributions.The marginal distribution of Y, that is fY(y)