       Re: Mathematical expectation

• To: mathgroup at smc.vnet.net
• Subject: [mg84131] Re: [mg84113] Mathematical expectation
• From: Carl Woll <carlw at wolfram.com>
• Date: Tue, 11 Dec 2007 06:06:14 -0500 (EST)
• References: <200712110140.UAA18271@smc.vnet.net>

```Benoît Frénay wrote:

>Hello,
>
>I'm new to Mathematica and I'm trying to define a generic mathematical
>expectation operator (called Esp) for i.i.d variables X[i] following the
>same distribution as the variable X (i.e. Esp[X[i]] == Esp[X]).  My code is
>
>Esp[X[_]] := Esp[X]
>Esp[X[_]^y_] := Esp[X^y]
>Esp[Esp[x_]] := Esp[x]
>Esp[Esp[x_]^y_] := Esp[x]^y
>Esp[x_ + y_] := Esp[x] + Esp[y]
>Esp[x_ y_] := Esp[x] Esp[y] /; FreeQ[x, y] && FreeQ[y, x]
>Esp[c_] := c /; FreeQ[c, X]
>Esp[c_ x_] := c Esp[x] /; NumberQ[c]
>
>But when I feed
> >>> In:= Esp[(X + X)^2]
>I obtain
> >>> Out= Esp[(X + X)^2]
>In fact, I obtain the correct answer only if I ask Mathematica to expand
>the inner term:
>In:= Esp[Expand[(X + X)^2]]
>Out= 2 Esp[X]^2 + 2 Esp[X^2]
>
>Where am I wrong?
>
>Thanks,
>
>Benoît Frénay
>
>
>
You don't have a rule for Esp[x_Plus^n_], so naturally nothing happens.
Perhaps you can add the rule:

Esp[x_Plus^n_] := Esp[Expand[x^n]]

Carl Woll
Wolfram Research

```

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