# solving a equation with integral

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Louis Liu on 22 Aug 2017
Edited: Louis Liu on 29 Aug 2017
Hello,
I write some code like this,
>> C = 1.00;
>> n = 38;
>> alpha = 0.05;
>> Cp = C+0.33;
>> fun=@(y)chi2cdf((n-1)*(3*Cp*sqrt(n)-y).^2/(9*n*czero^2),n-1).*(normpdf(y+3*(Cp-C)*sqrt(n))+normpdf(y-3*(Cp-C)*sqrt(n)));
>> solve(integral(fun,0,3*Cp*sqrt(n))-alpha,czero)
and get the main error message : Undefined function or variable 'czero'.
Does anyone can tell me how to solve this equation? Using symbolic variable or other method?
Thanks!

Walter Roberson on 22 Aug 2017
What you would like to do is roughly
C = 1.00;
n = 38;
alpha = 0.05;
Cp = C+0.33;
syms y czero real
fun = chi2cdf((n-1)*(3*Cp*sqrt(n)-y).^2/(9*n*czero^2),n-1).*(normpdf(y+3*(Cp-C)*sqrt(n))+normpdf(y-3*(Cp-C)*sqrt(n)));
solve( int(fun, y, 0,3*Cp*sqrt(n))-alpha, czero)
Unfortunately, chi2cdf does not expect symbolic values, and naively does a test on whether the input < 0 (because input values below 0 give an output of 0), but that test does not work on symbolic values. You can avoid the test by coding chi2pdf in terms of gampdf() but that only postpones the problem, as gampdf() has the same difficulty.
So, instead you have to work numerically:
C = 1.00;
n = 38;
alpha = 0.05;
Cp = C+0.33;
fun = @(y, czero) chi2cdf((n-1)*(3*Cp*sqrt(n)-y).^2/(9*n*czero^2),n-1).*(normpdf(y+3*(Cp-C)*sqrt(n))+normpdf(y-3*(Cp-C)*sqrt(n)));
czero_guess = 0;
czero_sol = fzero( @(czero) integral( @(y) fun(y, czero), 0, 3*Cp*sqrt(n) ) - alpha, czero_guess )
The output I am getting is -1.26021065783808
Louis Liu on 23 Aug 2017
Oh, you are right! Thanks!