Probability Distribution of X+Y and X/Y

jugantus · November 19th, 2010, 7:59 pm

Random variable X is distributed as N(a, b), and random variable Y is distributed as N(c, d). What is the distribution of (1) X+Y, (2) X-Y, (3) X*Y, (4) X/Y?

Hansi · November 19th, 2010, 9:40 pm

"If X1, X2 are two independent normal random variables, with means μ1, μ2 and standard deviations σ1, σ2, then their linear combination will also be normally distributed"So IIRC:1) N(a+c,b+d)2) N(a-c,b-d)3) Normal product distribution for zero means, not sure for non zero means. Someone?4) Cauchy distribution

BramJ · November 20th, 2010, 9:44 am

Make #2 from Hansi N(a-c,b+d). X ~ N(0,1) and Y~N(0,1) with X and Y independent doesn't imply X-Y ~ N(0,0) ...

kinderchocolate · November 23rd, 2010, 1:22 am

What should the differences be? Isn't differences just behave like addition? So if the point 2 is invalid, point 1 should also be invalid.

Traden4Alpha · November 23rd, 2010, 1:26 am

Assuming b and d are the variances, then the variance of the sum of the two will be b+d and the variance of the difference of the two will be b+d. If b & d are standard deviations, then the resultant standard deviation will be sqrt(b^2+d^2) in both cases.

DevonFangs · November 26th, 2010, 11:30 pm

I was thinking, maybe there's a way for a general binary operator, say # (# can be +,*,-,/).Given p_X and p_Y the PDF of the rv X and Y, the PDF of Z=X#Y isPlease share thoughts. EDIT: of course inside the delta I meant x#y but LaTeX with iPad is a big PITA