       Re: A problem with solving some nonlinear system

• To: mathgroup at smc.vnet.net
• Subject: [mg87319] Re: [mg87283] A problem with solving some nonlinear system
• From: Bob Hanlon <hanlonr at cox.net>
• Date: Mon, 7 Apr 2008 05:19:29 -0400 (EDT)

```dist = NormalDistribution[0, 1];

f[x_] = PDF[dist, x]

1/(E^(x^2/2)*Sqrt[2*Pi])

eqns1 = {
Integrate[f[z], {z, (c - 75)/10/Sqrt[n], Infinity}] == 0.05,
Integrate[f[z], {z, (c - 78)/10/Sqrt[n], Infinity}] == 0.9}

{(1/2)*Erfc[(c - 75)/(10*Sqrt*Sqrt[n])] == 0.05,
(1/2)*Erfc[(c - 78)/(10*Sqrt*Sqrt[n])] == 0.9}

FindRoot[eqns1, {{n, 1}, {c, 75}}]

{n->0.0105093,c->76.6862}

Off[Solve::ifun];

NSolve[eqns1, {n, c}][]

{n->0.0105093,c->76.6862}

Alternatively, use CDF directly rather than integrating the PDF

eqns2 = {1 - CDF[dist, (c - 75)/10/Sqrt[n]] == 0.05,
1 - CDF[dist, (c - 78)/10/Sqrt[n]] == 0.9}

{(1/2)*(-Erf[(c - 75)/(10*Sqrt*Sqrt[n])] - 1) + 1 == 0.05,
(1/2)*(-Erf[(c - 78)/(10*Sqrt*Sqrt[n])] - 1) + 1 == 0.9}

FindRoot[eqns2, {{n, 1}, {c, 75}}]

{n->0.0105093,c->76.6862}

NSolve[eqns2, {n, c}][]

{n->0.0105093,c->76.6862}

Bob Hanlon

---- Walkman <uvnarae at hotmail.com> wrote:
> Hi. This is the first posting in this board(in Englsigh?). So, if
> there is any punctuation error or misunderstood, please let me know.
>
> The problem itself is in the book 428p. "Introduction to Mathematical
> Statistics 6E"
>
> In solving this problem, I've got stuck with this practical problem.
>
> To find n and c such that
>
> Integrate[f,{z,(c-75)/10/sqrt(n),inf}] = .05
> Integrate[f,{z,(c-78)/10/sqrt(n),inf}] = .9
>
> where f = N(0,1); N -> Normal Distribution of which mean = 0 and
> variance = 1
>
> How can I solve this problem in mathematica? or any math-computational
> program? (e.g. R)
>

```

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