dstefan
06-16-2008, 12:46 PM
Chapter 4, Problem 1
Let X_1 = Z and X_2 = -Z be two independent random variables, where Z is the standard normal variable. Show that X_1 + X_2 is a normal variable of mean 0 and variance 2, i.e., X_1 + X_2 = \sqrt{2} Z.
Solution:
Recall that if X_1 and X_2 are independent normal random variables of mean and variance \mu_1 and \sigma_1^2, and \mu_2 and \sigma_2^2, respectively, then X_1 + X_2 is a normal variable of mean \mu_1 + \mu_2 and variance \sigma_1^2+\sigma_2^2, i.e.,
X_1 + X_2 ~=~ \mu_1 + \mu_2 ~+~ \sqrt{\sigma_1^2 + \sigma_2^2} Z
For X_1 = Z and X_2 = -Z, it follows that \mu_1 = \mu_2 = 0 and \sigma_1 = \sigma_2 = 1. We conclude that
E[X] = \mu_1 + \mu_2 = 0; \mbox{var}(X) = \sigma_1^2 + \sigma_2^2 = 2 and therefore X ~=~ X_1 + X_2 ~=~ \sqrt{2} Z
Let X_1 = Z and X_2 = -Z be two independent random variables, where Z is the standard normal variable. Show that X_1 + X_2 is a normal variable of mean 0 and variance 2, i.e., X_1 + X_2 = \sqrt{2} Z.
Solution:
Recall that if X_1 and X_2 are independent normal random variables of mean and variance \mu_1 and \sigma_1^2, and \mu_2 and \sigma_2^2, respectively, then X_1 + X_2 is a normal variable of mean \mu_1 + \mu_2 and variance \sigma_1^2+\sigma_2^2, i.e.,
X_1 + X_2 ~=~ \mu_1 + \mu_2 ~+~ \sqrt{\sigma_1^2 + \sigma_2^2} Z
For X_1 = Z and X_2 = -Z, it follows that \mu_1 = \mu_2 = 0 and \sigma_1 = \sigma_2 = 1. We conclude that
E[X] = \mu_1 + \mu_2 = 0; \mbox{var}(X) = \sigma_1^2 + \sigma_2^2 = 2 and therefore X ~=~ X_1 + X_2 ~=~ \sqrt{2} Z