variance of product of 2 independent variables

• To: mathgroup at smc.vnet.net
• Subject: [mg87030] variance of product of 2 independent variables
• From: "Dankwort, Rudolf C" <Rudolf.Dankwort at Honeywell.com>
• Date: Sat, 29 Mar 2008 04:25:10 -0500 (EST)

```Hello Ben -

I have a question about the subject matter.  To review, you sent Frank
Brand the following:

<<If a and b were completly uncorrelated (not even non-linear
correlations among them),
then you can compute the variance of their product quite easily

v(ab) := < a^2b^2 > - < ab >^2 = <a^2><b^2> - <a>^2<b>^2 = v(a)<b> +
v(b)<a> + v(a)v(b);
v(a)=<a^2>-<a>^2, v(b)=<b^2>-<b>^2

here v(.) denotes variance, <.> denotes mean.
Note that we do not have to assume normal distributions for a and b,
essential is that their are uncorrelated, hence the means of products
factor into products of means.>>

If <a> = 1000 and <b> = 0.001, and v(a) = 100 and v(b) = 1e-10 (in other
words, both a and b have 1% standard deviations), then I compute v(ab) =
100*0.001 + 1e-10*1000 + 100*1e-10 ~ 0.1
which is obviously wrong (<ab> = 1.000 and std deviation would be
sqrt(0.1) = 0.31.

Help!  Da stimmt was nicht!

Rudy dankwort
Phoenix AZ USA

```

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