Re: Linear combinations of Expectation of EmpiricalDistribution

• To: mathgroup at smc.vnet.net
• Subject: [mg128159] Re: Linear combinations of Expectation of EmpiricalDistribution
• From: Bob Hanlon <hanlonr357 at gmail.com>
• Date: Wed, 19 Sep 2012 04:59:59 -0400 (EDT)
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• References: <20120918074019.14F1B6847@smc.vnet.net>

```Expectation[x,
x \[Distributed] EmpiricalDistribution[{0, 1, 2}]]

1

Expectation[y,
y \[Distributed] EmpiricalDistribution[{0, 10, 20}]]

10

Expectation[x + y,
{x, y} \[Distributed] EmpiricalDistribution[
Thread[{{0, 1, 2}, {0, 10, 20}}]]]

11

Expectation[a*x + b*y,
{x, y} \[Distributed] EmpiricalDistribution[
Thread[{{0, 1, 2}, {0, 10, 20}}]]]

a + 10 b

Bob Hanlon

On Tue, Sep 18, 2012 at 3:40 AM, Clemens Fruhwirth
<clemens at endorphin.org> wrote:
> Dear MathGroup,
>
> The expectation for any linear combination of random values is the
> linear combination of the respective expectations of the random
> values, such as
>
> E(aX+bY) = aE(X) + bE(Y)
>
> I wonder why Mathematica can't resolve this rule when
> EmpiricalDistributions come into play. For instance,
>
> Expectation[
>  x + y, {x \[Distributed] EmpiricalDistribution[{0, 1, 2}],
>   y \[Distributed] EmpiricalDistribution[{0, 10, 20}]}]
>
> I came up with:
>
> ExpectationX[
>   a_ + b_, {x_ \[Distributed] xdist_, y_ \[Distributed] ydist_}] :=
>  a + b //. {x -> Expectation[x, x \[Distributed] xdist],
>    y -> Expectation[y, y \[Distributed] ydist]}
>
> to resolve at least simple addition. But before I put more work into that:
>
> * Am I missing an assumption here or some syntax? Or is this rule just
> not built into Mathematica?
>
> * Is that in general the right approach to extend Mathematica?
>
> Thanks!
> --
> Fruhwirth Clemens http://clemens.endorphin.org
>

```

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