Question two (30) 0 Fx(z)otherwise = e-x, what is the pdf of Y Find out E(1-Y2) Say Z = λΥ.λ > 0,...
4. Suppose that X and Y have the following joint PDF: e-(z+y) fx,Y(x,y) = :x>0, y > 0 : otherwise Use the CDF method to find and identify the distribution of WX
Problem 2 - Three Continuous Random Variables Suppose X,Y,Z have joint pdf given by fx,YZ(xgz) = k xyz if 0 S$ 1,0 rS 1,0 25 1 ) and fxyZ(x,y,z) = 0, otherwise. (a) Find k so that fxyz(x.yz) is a genuine probability density function. (b) Are X,Y,Z independent? (c) Find PXs 1/2, Y s 1/3, Z s1/4). (d) Find the marginal pdf fxy(x.y). (e) Find the marginal pdf fx(x).
Problem 2 - Three Continuous Random Variables Suppose X,Y,Z have joint...
1. Consider a pair of random variables (X, Y) with joint PDF fx,y(x, y) 0, otherwise. (a) 1 pt - Find the marginal PDF of X and the marginal PDF of Y. (b) 0.5 pt - Are X and Y independent? Why? (e) 0.5 pt - Compute the mean of X and the mean of Y.
X and Y are random variables with the joint PDF fx.^(t,y)-65536 0 otherwise. (a) What is the marginal PDFfx(x)? ㄑㄨ 8 5xA4/65536 fx(x) 0 otherwise (b) What is the marginal PDF fy(v)? (5 * 843)/(3*655 0 〈y〈 64 fy(y) = 0 otherwise
Question 5 15 marks] Let X be a random variable with pdf -{ fx(z) = - 0<r<1 (1) 0 :otherwise, Xa, n>2, be iid. random variables with pdf where 0> 0. Let X. X2.... given by (1) (a) Let Ylog X, where X has pdf given by (1). Show that the pdf of Y is Be- otherwise, (b) Show that the log-likelihood given the X, is = n log0+ (0- 1)log X (0 X) Hence show that the maximum likelihood...
Suppose X and Y are two continuous random variables with probability density functions: fx(x)1 for 1<x2, fx(x) 0 otherwise, and fr (v) 3e3y for y>0, fr (y) 0 otherwise. a) Suppose X and Y are independent, is Z-X+ Y"memoryless"? Justify your answer. b) Suppose that the conditional expected value satisfies E(Y X)-X. Find Cov0), and El(Y-X) expX)].
Suppose X and Y are two continuous random variables with probability density functions: fx(x)1 for 10, fr (y) 0 otherwise. a) Suppose X...
Problem 5. The joint density of X and Y is given by e" (z+y) fx.-otherwise. İf 0 < x < oo, 0 < y < 00, Consider the random variable Z-; a) Find the cumulative distribution function of Z b) What is the probability density function of Z?
The random variables X and Y have joint PDF fX,Y(x,y) = {12x2y 0<=x<=c; 0 <= y <= 3 { 0 otherwise (a) FInd the value of C (b) Find the PDF fW(w) where W = X / Y (c) Find the PDF fZ(z) where Z = min(X,Y)
Q3. Suppose that X, Y have joint pdf a for x2 + y2 0 otherwise. 1. fxy(x, y)- (a) Find the value of a so that fxy(x, y) is a valid pdf. b) Find the marginal pdf for X Hint: It is helpful to sketch the region of the ry-plane where the pdf is non-zero
0 〈 y 〈 x2く1· Consider two rvs X and Y with joint pdf f(x,y) = k-y, (a) Sketch the region in two dimensions where fx,y) is positive. Then find the constant k and sketch ) in three imesions Then find the constant k and sketch f(r.y) in three dimensions (b) Find and sketch the marginal pdf fx), the conditional pdf(x1/2) and the conditional cdf FO11/2). Find P(X〈Y! Y〉 1/2), E(XİY=1/2) and E(XIY〉l/2). (c) What is the correlation between X...