standard normal random varible?

Question

let X1 and X2 be two independent standard normal random variable. Let Y1=X1+X2 and Y2=X1/X2

a. find the joint probability density function of Y1 and Y2
b. find the marginal probability density function of Y1

Answer 1

I'll help you with answers for Y1. Since X1 and X2 are independtly distributed N(0, 1) variables (ie. normal random variables with mean 0 and variation 1), Y1 = X1 + X2 is a normal random variable of mean 0 + 0 = 0, and variation 1 + 1 = 2. Thus the pdf of Y1 is that of an N(0, 2) variable:

f(x) = (1/2√π) exp(- x²/4),

where exp() denotes the exponential function. This answer a.

For part b., observe that since X1 and X2 are independent, the marginal distribution of each wrt the other is simply the same as the pdf for each. In each case this is the pdf for an N(0, 1) distribution, ie.

f(x) = (1/√(2π)) exp(- x²/2).