How can I solve an integral problem included of cumulative distribution function of chi-square distribution?

Excuse me, does anyone know how to solve an integral problem included of cumulative distribution function of chi-square distribution? For example, there is a question like the attached picture:

I want to use the int() function to solve it, however, there are two probability distribution in it, and i don't know how to deal with it.

I hope you can tell me how to solve it like using which function or method instead of giving me the correct answer. Thanks! c = 1.00, n = 100, c = 1.15, Cp = 1.12

[Torsten]

https://de.mathworks.com/help/matlab/ref/integral.html

together with

https://de.mathworks.com/help/stats/chi2cdf.html
https://de.mathworks.com/help/stats/normpdf.html

should work.

Best wishes

Torsten.

[Louis Liu]

Hello Torsten, Actually I also find the function chi2cdf, however, when I want to use it like the attached photo,

it couldn't work. I look back the example of documentation of chi2cdf, it seems like I couldn't use eq1 variable as x in the example and then to solve the integral. Can you give me more instructions? Thanks!

@Torsten

Answer 1

[Torsten]

fun=@(y)chi2cdf((n-1)*(3*Cp*sqrt(n)-y).^2/(9*n*cstar^2),n-1).*(normpdf(y+3*(Cp-C)*sqrt(n))+normpdf(y-3*(Cp-C)*sqrt(n)));
result=integral(fun,0,3*Cp*sqrt(n));

Of course, all the constants have to be given numerical values before calling "integral".

Don't use "syms" anywhere.

Best wishes

Torsten.

[Louis Liu]

Thanks, I am so silly...

[Louis Liu]

@Torsten Hello, Can you give me a brief reason why shouldn't I use "syms" anywhere? Thanks!

[Torsten]

Because "integral" can not handle symbolic variables and "int" is inappropriate to use because it won't be able to calculate an analytic antiderivative for your integrand (I guess).

Best wishes

Torsten.

[Louis Liu]

if true
>> 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)
end

@Torsten Hello, can I ask you a similar question? I want to solve a equation like attached photo. So I write some code above.

I guess the main problem is "czero". The value of czero is what I want to know, so I couldn't assign the initial value for czero and maybe couldn't run the integral function. I also try to use >>syms czero; however, it still couldn't work like you tell me in the last time. Can you give me some advice? Thank you so much!

[Torsten]

Try

C = 1.00;
n = 38;
alpha = 0.05;
Cp = C + 0.33;
fun = @(czero,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)));
czero0 = 1.0;
sol = fzero(@(czero)integral(@(y)fun(czero,y)-alpha,0,3*Cp*sqrt(n),'ArrayValued',true),czero0);

Best wishes

Torsten.

[Louis Liu]

@Torsten Thanks, I will try it!