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Re: A nasty 2x2 system of equations?


Bruce Colletti 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}]
> 

<< Statistics`ContinuousDistributions`
X = BetaDistribution[a1, b1];
Y = BetaDistribution[a2, b2];
Z = BetaDistribution[a3, b3];
eq1 = Mean[Z] - c*Mean[X] - (1 - c)*Mean[Y] // Together // Numerator;
eq2 = Variance[Z] - c^2*Variance[X] - (1 - c)^2*Variance[Y] //
  Together // Numerator;
Solve[{eq1 == 0, eq2 == 0}, {a3, b3}];

but don't expect a useful expression except with real number input.

--

Roland Franzius


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