Re: A nasty 2x2 system of equations?

• To: mathgroup at smc.vnet.net
• Subject: [mg54137] Re: [mg54098] A nasty 2x2 system of equations?
• From: DrBob <drbob at bigfoot.com>
• Date: Fri, 11 Feb 2005 03:34:10 -0500 (EST)
• References: <200502100747.CAA16654@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

```It's a nasty problem, but here's what appears to be a solution:

meanEQ = Mean[Z] == c*Mean[X] + (1 - c)*Mean[Y];
varianceEQ = Variance[Z] == c^2*Variance[X] + (1 - c)^2*Variance[Y];

Solving the first equation:

Off[Solve::"svars"]
a3Rule = First@Solve[meanEQ, {a3, b3}]

{a3 -> -((b3*(a1*a2 + a2*b1 -
a2*b1*c + a1*b2*c))/
((-a1)*b2 - b1*b2 -
a2*b1*c + a1*b2*c))}

(checking)

meanEQ/.a3Rule//Simplify

True

Applying that to the other equation and solving:

Solve[varianceEQ /. a3Rule, {b3}]

{{b3 -> (a1^2*a2*b2^2 +
a1^3*a2*b2^2 + 2*a1*a2*b1*
b2^2 + 3*a1^2*a2*b1*
b2^2 + a2*b1^2*b2^2 +
3*a1*a2*b1^2*b2^2 +
a2*b1^3*b2^2 + a1*a2*b1*b2*
c + a1^2*a2*b1*b2*c +
2*a1*a2^2*b1*b2*c +
2*a1^2*a2^2*b1*b2*c +
a2*b1^2*b2*c + 2*a1*a2*
b1^2*b2*c + 2*a2^2*b1^2*
b2*c + 4*a1*a2^2*b1^2*b2*
c + a2*b1^3*b2*c +
2*a2^2*b1^3*b2*c +
a1^2*b2^2*c + a1^3*b2^2*
c - 2*a1^2*a2*b2^2*c -
2*a1^3*a2*b2^2*c +
a1*b1*b2^2*c + 2*a1^2*b1*
b2^2*c - 3*a1*a2*b1*b2^2*
c - 5*a1^2*a2*b1*b2^2*c +
a1*b1^2*b2^2*c -
a2*b1^2*b2^2*c -
4*a1*a2*b1^2*b2^2*c -
a2*b1^3*b2^2*c +
a1^2*b2^3*c + a1^3*b2^3*
c + a1*b1*b2^3*c +
2*a1^2*b1*b2^3*c +
a1*b1^2*b2^3*c +
a2^2*b1^2*c^2 + a1*a2^2*
b1^2*c^2 + a2^3*b1^2*
c^2 + a1*a2^3*b1^2*c^2 +
a2^2*b1^3*c^2 + a2^3*b1^3*
c^2 - a1*a2*b1*b2*c^2 -
3*a1*a2^2*b1*b2*c^2 -
2*a1^2*a2^2*b1*b2*c^2 -
a2*b1^2*b2*c^2 -
a1*a2*b1^2*b2*c^2 -
2*a2^2*b1^2*b2*c^2 -
4*a1*a2^2*b1^2*b2*c^2 -
a2*b1^3*b2*c^2 -
2*a2^2*b1^3*b2*c^2 -
2*a1^2*b2^2*c^2 -
2*a1^3*b2^2*c^2 +
a1^2*a2*b2^2*c^2 +
a1^3*a2*b2^2*c^2 -
a1*b1*b2^2*c^2 -
2*a1^2*b1*b2^2*c^2 +
2*a1*a2*b1*b2^2*c^2 +
5*a1^2*a2*b1*b2^2*c^2 +
4*a1*a2*b1^2*b2^2*c^2 -
2*a1^2*b2^3*c^2 -
2*a1^3*b2^3*c^2 -
a1*b1*b2^3*c^2 -
2*a1^2*b1*b2^3*c^2 -
a2^2*b1^2*c^3 - a2^3*b1^2*
c^3 - a2^2*b1^3*c^3 -
a2^3*b1^3*c^3 - a1^2*a2*b1*
b2*c^3 + a1*a2^2*b1*b2*
c^3 + a1*a2*b1^2*b2*c^3 +
3*a1*a2^2*b1^2*b2*c^3 +
a1^2*b2^2*c^3 + a1^3*b2^2*
c^3 - a1*a2*b1*b2^2*c^3 -
3*a1^2*a2*b1*b2^2*c^3 +
a1^2*b2^3*c^3 + a1^3*b2^3*
c^3)/(a1^2*a2*b2 +
a1^3*a2*b2 + 2*a1*a2*b1*
b2 + 3*a1^2*a2*b1*b2 +
a2*b1^2*b2 + 3*a1*a2*b1^2*
b2 + a2*b1^3*b2 -
2*a1^2*a2*b2*c -
2*a1^3*a2*b2*c -
4*a1*a2*b1*b2*c -
6*a1^2*a2*b1*b2*c -
2*a2*b1^2*b2*c -
6*a1*a2*b1^2*b2*c -
2*a2*b1^3*b2*c +
a1*a2^2*b1*c^2 +
a1*a2^3*b1*c^2 +
a1^2*a2*b2*c^2 +
a1^3*a2*b2*c^2 +
4*a1*a2*b1*b2*c^2 +
3*a1^2*a2*b1*b2*c^2 +
3*a1*a2^2*b1*b2*c^2 +
a2*b1^2*b2*c^2 +
3*a1*a2*b1^2*b2*c^2 +
a2*b1^3*b2*c^2 +
a1*b1*b2^2*c^2 +
3*a1*a2*b1*b2^2*c^2 +
a1*b1*b2^3*c^2)}}

Yikes!! Solving in the other order (varianceEQ first) doesn't work so well.

Bobby

On Thu, 10 Feb 2005 02:47:14 -0500 (EST), Bruce Colletti <vze269bv at verizon.net> wrote:

> Re Mathematica 5.1.
>
> The code below keeps running without getting an answer.  Is the code flawed or is this a really nasty system of 2-equations in 2-unknowns?
>
> Here's background:  X and Y are independent beta-distributed random variables and Z is a convex combination of X and Y, i.e., Z = cX + (1 - c)Y.
>
> Although Z need not be beta-distributed, let's pretend it is and in turn, solve for its parms (a3 and b3) in terms of c and the known parms of X and Y.
>
> Thanks.
>
> Bruce
>
> -------------
>
>
>
> Solve[{Mean[Z] == c*Mean[X] + (1 - c)*Mean[Y],
>    Variance[Z] == c^2*Variance[X] + (1 - c)^2*Variance[Y]}, {a3, b3}]
>
>
>
>

--
DrBob at bigfoot.com
www.eclecticdreams.net

```

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