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This is a Statistics problem.  Are the random variables Z1+Z2 and Z1-Z2 the same? Do they have the same distribution?

This is a Statistics problem.  Are the random variables Z1+Z2 and Z1-Z2 the same? Do they have the same distribution?

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
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This is a Statistics problem. 

Are the random variables Z1+Z2 and Z1-Z2 the same? Do they have the same distribution? 

Estimate the pdf of Z1 + Z2 where Z, and Z2 are independent standard normal random variables. z1 <- rnorm(10000) 22 <- rnorm(10000) plot(density(Z1 + 22), main = "Sum of two standard normal rvs", xlab = expression (Z[1] + Z[2]) curve(dnorm(x, 0, sąrt(2)), add = TRUE, col = "red") Sum of two standard normal rvs -6 -4 -2 4. 6. Z, +Z2 In the plot, the pdf for a Norm(0, v2) rv is included. It appears that the sum of two standard normal random variables is again a normal random variable. The standard deviation is v2 since Var(Z1 + Z2) = Var(Z1) + Var(Z2) = 2 by Theorem 3.10. Try It Yourself For Z1,Zz independent standard normal, estimate the pdf of Z - Zz and see that it is also Norm(0, v2). Density 0.00 0.10 0.20 0.30
Transcribed Image Text:Estimate the pdf of Z1 + Z2 where Z, and Z2 are independent standard normal random variables. z1 <- rnorm(10000) 22 <- rnorm(10000) plot(density(Z1 + 22), main = "Sum of two standard normal rvs", xlab = expression (Z[1] + Z[2]) curve(dnorm(x, 0, sąrt(2)), add = TRUE, col = "red") Sum of two standard normal rvs -6 -4 -2 4. 6. Z, +Z2 In the plot, the pdf for a Norm(0, v2) rv is included. It appears that the sum of two standard normal random variables is again a normal random variable. The standard deviation is v2 since Var(Z1 + Z2) = Var(Z1) + Var(Z2) = 2 by Theorem 3.10. Try It Yourself For Z1,Zz independent standard normal, estimate the pdf of Z - Zz and see that it is also Norm(0, v2). Density 0.00 0.10 0.20 0.30
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