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MathGroup Archive 2005

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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>
  • Reply-to: drbob at bigfoot.com
  • Sender: owner-wri-mathgroup at wolfram.com

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

X = BetaDistribution[a1, b1];
Y = BetaDistribution[a2, b2];
Z = BetaDistribution[a3, b3];
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
>
> -------------
>
>
> X = BetaDistribution[a1, b1];
> Y = BetaDistribution[a2, b2];
> Z = BetaDistribution[a3, b3];
>
> 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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