11. Let X and Y have the joint density12--(x+y)/0for 0 < x < y <∞ and zero elsewhere02Find a such that aY is an unbiased statistic for 0 and find its variance. (Note that the rolesof X and Y are not symmetric due to the condition that 0 < x < y < ∞.)

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Answered: 11. Let X and Y have the joint density…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

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Problem 1P

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We have the following information about the random variables X and Y: |x = 1, |y = −1.5 , ox = 0.25, c} = 1. o Calculate the variance of Z = 3X + 5Y, a) when the coefficient of correlation is p(X, Y) = 0.46: 0²77 34.15 Z b) when X and Y are independent random variables: 0²2 27.75 X
Suppose that the response y is generated by y = f(x) + €, where e is a zero-mean Gaussian noise with variance 1. a) Suppose that f(x) = x. Randomly generate 10 x's and generate the corresponding y's; you need to generate two random numbers (i.e., x and e for each of the 10 points). Fit the data with linear regression and plot the scatter points. b) Suppose that f(x) = x². Randomly generate 10 (x, y) pairs. Fit the data with linear regression and plot the scatter points. c) Suppose that f(x) = 1/x. Randomly generate 10 (x, y) pairs. Fit the data with linear regression and plot the scatter points.
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8. Let the random variable X have the pdf 2 x2 fx (x) = exp %3D - V2n 2 Find the mean and the variance of X. Hint: Compute E (X) directly and E (X²) by comparing the integral with the integral representing the variance of a random variable that is N(0,1). i DCO 04 < (X - 5)2 < 38.4).
Let X, Y, Z be the three assets, with VAR(X) = VAR(Y) = VAR(Z) = 1. Z is independent from X and Y , and COV(X, Y) = 0.2 .Assuming no short selling, what is the portfolio weight on Z that gives the smallest variance? (Hint: since Z is independent with X and Y, you may express wx and wy as a function of 1- wz , and then find wz.)
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2

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11. Let X and Y have the joint density 1 -e-(x+y)/0 for 0 < x < y < o and zero elsewhere 02 Find a such that aY is an unbiased statistic for 0 and find its variance. (Note that the roles of X and Y are not symmetric due to the condition that 0 < x < y < ∞.)