Finding unknown parameters using Mathematica

• To: mathgroup at smc.vnet.net
• Subject: [mg74722] Finding unknown parameters using Mathematica
• Date: Mon, 2 Apr 2007 06:55:58 -0400 (EDT)

```Dear group members,

I'm very new to mathematica and trying to solve a set of non-linear system =
of equations to find the unknown parameters for a bivariate distribution. =
I've 5 unknown parameters (i.e. b1,b2,r1,r2,p) and 5 set of equations.  I =
tried to get the general solution , but could not. I don't know how to =
solve these nonlinear equations to get the unknown parameters. And also =
not aware how to give initial value in the solve function or any other =
function (e.g all parameters b1,b2,r1,r2,p if I give initial boundary =
value =1).
In the below codes, x1 and x2 are 2 variables (e.g. data from 2 quality =
characteristics; stress and strain etc. etc.)

Any comments / suggestions how to solve these equation for b1,b2,r1,r2,p =
where as putting intial value for all these unknown parameters =1

===================
n=4
x1={1,2,3,4}
x2={1.7,3.8,4.9,4.6}

4

{1,2,3,4}

{1.7,3.8,4.9,4.6}

\!\(\*
RowBox[{
RowBox[{
StyleBox["L",
FontSize->10],
StyleBox["=",
FontSize->14],
StyleBox[\(n*Log[p] + n*Log[p + 1] + n*
Log[b1] + n*Log[r1] + n*Log[b2] + n*Log[r2] + \((b1 - 1)\) =
\(=E2=88=91\+\(j =
1\)\%n Log[x1[\([
j]\)]]\) + \((b2 -
1)\) \(=E2=88=91\+\(j = 1\)\%n Log[
x2[\([j]\)]]\) - \((p + 2)\) \(=E2=88=91\+\(j = 1\)\%n Log[1 +
r1*\((x1[\([j]\)]^b1)\) + r2*\((x2[\([j]\)]^b2)\)]\)\),
FontSize->14,
FontColor->RGBColor[1, 0, 0]]}], "\[IndentingNewLine]",
StyleBox[" ",
FontSize->18]}]\)

\!\(4.980920826406141`\ \((\(-1\) + b2)\) + \((\(-1\) + b1)\)\ \((
Log[2] + Log[3] + Log[
4])\) + 4\ Log[b1] + 4\ Log[b2] + 4\ Log[
p] + 4\ Log[1 + p] + 4\ Log[r1] + 4\ Log[r2] - \((2 + p)\)\ \((
Log[1 + r1 + 1.7`\^b2\
r2] + Log[1 + 2\^b1\ r1 + 3.8`\^b2\ r2] +
Log[1 + 4\^b1\ r1 + 4.6`\^b2\ r2] + Log[1 + 3\^b1\ r1 + =
4.9`\^b2\
r2])\)\)

Eqn1=D[L,r1]\[Equal]0

\!\(4\/r1 - \((2 +
p)\)\ \((1\/\(1 + r1 + 1.7`\^b2\
r2\) + 2\^b1\/\(1 + 2\^b1\ r1 + 3.8`\^b2\ r2\) + 4\^b1\/\(1 +
4\^b1\ r1 + 4.6`\^b2\ r2\) + 3\^b1\/\(1 + 3\^b1\ r1 + =
4.9`\^b2\ \
r2\))\) \[Equal] 0\)

Eqn2=D[L,r2]\[Equal]0

\!\(4\/r2 - \((2 +
p)\)\ \((1.7`\^b2\/\(1 +
r1 + 1.7`\^b2\ r2\) + 3.8`\^b2\/\(1 + 2\^b1\ r1 + 3.8`\^b2\ r2\) =
+ \
4.6`\^b2\/\(1 +
4\^b1\ r1 + 4.6`\^b2\ r2\) + 4.9`\^b2\/\(1 + 3\^b1\ r1 + =
4.9`\^b2\
\ r2\))\) \[Equal] 0\)

Eqn3=D[L,b1]\[Equal]0

\!\(4\/b1 + Log[2] + Log[3] + Log[4] - \((
2 + p)\)\ \((\(2\^b1\ r1\
Log[2]\)\/\(1 + 2\^b1\ r1 + 3.8`\^b2\
r2\) + \(3\^b1\ r1\ Log[3]\)\/\(1 + 3\^b1\ r1 + 4.9`\^b2\ r2\) + =
\
\(4\^b1\ r1\ Log[4]\)\/\(1 + 4\^b1\ r1 + 4.6`\^b2\ r2\))\) \[Equal] 0\)

Eqn4=D[L,b2]\[Equal]0

\!\(\(\(4.980920826406141`\)\(\[InvisibleSpace]\)\) +
4\/b2 - \((2 + p)\)\ \((\(0.5306282510621704`\ 1.7`\^b2\ r2\)\/\(1 =
+
r1 + 1.7`\^b2\
r2\) + \(1.33500106673234`\ 3.8`\^b2\ r2\)\/\(1 + 2\^b1\ r1 + =
\
3.8`\^b2\ r2\) + \(1.5260563034950492`\ 4.6`\^b2\ r2\)\/\(1 + 4\^b1\ r1 + =
\
4.6`\^b2\ r2\) + \(1.589235205116581`\ 4.9`\^b2\ r2\)\/\(1 + 3\^b1\ r1 + =
4.9`\
\^b2\ r2\))\) \[Equal] 0\)

Eqn5=D[L,p]\[Equal]0

\!\(4\/p + 4\/\(1 + p\) - Log[1 + r1 + 1.7`\^b2\
r2] - Log[1 + 2\^b1\ r1 + 3.8`\^b2\ r2] - Log[1 + 4\^b1\
r1 + 4.6`\^b2\ r2] - Log[1 + 3\^b1\ r1 + 4.9`\^b2\ r2] \[Equal] 0\)

Solve[{Eqn1,Eqn2,Eqn3,Eqn4,Eqn5},{r1,r2,b1,b2,p}]

```

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