Variance of AxB?

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3 total messages Started by KUN YU Thu, 01 Oct 1992 21:12

#614

Author: KUN YU
Date: Thu, 01 Oct 1992 21:12

16 lines
888 bytes

Suppose A and B are vectors of random numbers. Each has multivariate normal 
distribution with 0 mean.
 A(1,n), ~N(sigma1, 0), i.e., each element in A has variance sigma1, mean 0.
 B(n,1), ~N(sigma2, 0)

What is the variance of $A {\times} B$?


-- 



                                                          _                  
ky04@lehigh.edu                                         / _ \
KYu, BioProcessing Institute                          /\ _|_ /\
111 Lehigh University                                    /|\

Re: Variance of AxB?

#629

Author: r hardin
Date: Sat, 03 Oct 1992 01:16

16 lines
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I seem to remember a handy trick for normal distributed numbers,
the expected value of any product of them   <abcd> say  is

<abcd> = <ab> <cd> + <ac> <bd> + <ad> <bc>

ie., just write every unique product of pairs

which works for any order of product.

From which, if I have remembered it correctly of course, you can
compute any variance.

Perhaps someone can verify the rule, which I am remembering from
a distance of perhaps 20 years.

Re: Variance of AxB?

#630

Author: Herman Rubin
Date: Sat, 03 Oct 1992 14:42

29 lines
1477 bytes

In article <23788@alice.att.com> rhh@alice.att.com (r hardin) writes:

>I seem to remember a handy trick for normal distributed numbers,
>the expected value of any product of them   <abcd> say  is

><abcd> = <ab> <cd> + <ac> <bd> + <ad> <bc>

>ie., just write every unique product of pairs

>which works for any order of product.

>From which, if I have remembered it correctly of course, you can
>compute any variance.

>Perhaps someone can verify the rule, which I am remembering from
>a distance of perhaps 20 years.

The rule is correct if the random variables have mean 0, and is
easily obtained by differentiating the moment generating function
the appropriate number of times and seeing which terms are not 0.

It is not valid if the means are not 0.

-- 
Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399
Phone: (317)494-6054
hrubin@snap.stat.purdue.edu (Internet, bitnet)  
{purdue,pur-ee}!snap.stat!hrubin(UUCP)

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