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 FORUMS > Student Forum < refresh >
 Topic Title: Probability Distribution of X+Y and X/Y Created On Fri Nov 19, 10 08:59 PM Topic View: Branch View Threaded (All Messages) Threaded (Single Messages) Linear

jugantus
Junior Member

Posts: 5
Joined: Oct 2010

Fri Nov 19, 10 08:59 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?

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Hansi
Senior Member

Posts: 2989
Joined: Jan 2010

Fri Nov 19, 10 10:40 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
Senior Member

Posts: 304
Joined: Jan 2006

Sat Nov 20, 10 10:44 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
Member

Posts: 26
Joined: Apr 2010

Tue Nov 23, 10 02:22 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.

Senior Member

Posts: 15834
Joined: Sep 2002

Tue Nov 23, 10 02:26 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.

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"It often happens that a player carries out a deep and complicated calculation, but fails to spot something elementary right at the first move." -- grandmaster Alexander Kotov --inscribed on gift chess sets given by Amaranth hedge fund.

DevonFangs
Senior Member

Posts: 2913
Joined: Nov 2009

Sat Nov 27, 10 12:30 AM

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 is

EDIT: of course inside the delta I meant x#y but LaTeX with iPad is a big PITA

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Consideration of various properties of the log-theta-lattice led naturally to the establishment, in the third paper of the series, of multiradial algorithms for constructing "splitting monoids of LGP-monoids".

Edited: Sat Nov 27, 10 at 12:32 AM by DevonFangs