       Re: Use of Mathematica with Rule-based Equation Derivations

• To: mathgroup at smc.vnet.net
• Subject: [mg61941] Re: [mg61914] Use of Mathematica with Rule-based Equation Derivations
• From: Daniel Lichtblau <danl at wolfram.com>
• Date: Sat, 5 Nov 2005 01:52:31 -0500 (EST)
• References: <200511041011.FAA14913@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

```mmorriss at gcn.ou.edu wrote:
> Hi All - I'm a many-year user of mathematica, but have always had one
> particular problem with it that I may have just simply missed reading
>
> Mathematica Version: 5.2
>
> Problem: I would like to develop a set of re-rewite rules to apply to the
> Expected Value operator.  For example:
>
> E[a x] = a E[x]  a -> constant, x -> variable
> E[b + f[x]] = b + E[f[x]] , etc.
>
> The issue is how does one using Mathematica distinguish a 'constant
> variable (i.e. a and b)' from a variable 'variable' (i.e. 'x')? The head
> of a, b and x is 'Symbol' and neither a, b nor x contain a number so I
> can't use a_?NumberQ to identify it as a constant.
>
> This actually goes to the wider question of how does one use Mathematica
> for symbolic derviations where numbers are not actually substituted in the
> derviation?
>
> E.g. E[a + E[b x]] /. Rule2
>    out= a + b E[x] etc.
>
> Thanks all - Mark Morrissey
> University of Oklahoma
>

One reasonable approach would be to use an explicit head to denote
random variables (as opposed to scalares). Pulling out constant factors
could then be done as below.

expectation := 1

expectation[a_+b_] := expectation[a]+expectation[b]

expectation[a_*b_.] /; FreeQ[a,randomVariable] := a*expectation[b]

In:= InputForm[expectation[a+c*randomVariable[b]^2]]

Out//InputForm= a + c*expectation[randomVariable[b]^2]

A standard recommendation is that you not call the operator E.

Daniel Lichtblau
Wolfram Research

```

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