solving a equation with integral

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!

Answer 1

[Walter Roberson]

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]

@Walter Roberson Thanks! I will try it and can you tell me why do you assign crezo_guess as value 0? Because the right answer didn't have a minus sign and if I change the initial value of crezo_guess to 1, I can get the positive sign. Thanks!

[Walter Roberson]

I had to choose something for the guess, and I had no idea of the value that would be found.

The only place your equation has czero is in the form czero^2 so your equation cannot tell the difference between negative and positive czero

[Louis Liu]

Oh, you are right! Thanks!