Integral of a product between a function and a Cumulative Normal Distribution

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Javier 2013-3-26

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Hi.

I need to compute an integral of a function that consist in the product between a Cumulative distribution of a Normal, and another term.

I tried using the command int, but in that case I was not able to execute the normcdf command.

Matlab said: CNaux2=normcdf(aux5) Error using NaN Trailing string input must be 'single' or 'double'.

Instead, i tried using the quad command. In that case, the normcdf could be defined as a function handle, but then i coul not excute the quad command. In this case matlab said

Error using * Inner matrix dimensions must agree.

And i don´t understand why, because i don´t have any matrix here, i have only a product of functions of one variable.

Could you please help me?

Thanks! Javier

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Javier 2013-3-27

What i have to do is to compute a definite integral for the function:

N(a(t))*b(t)*C(d(t)) dt

Where N(a(t)) is a normal probability function valued at the value a, which is in fact a function that depends on t. C(d(t)) is a cumulative normal probability function valued d, which is also a function that depends on t. And b(t) is another function of t.

I have tried to define first the functions a(t) b(t) and c(t) as function handles. Then i used the commandsnormcdf and normpdf to compute the normal cumulative and probability function on this function (i think this way i obtain also a function handle that depends on t, but i am not sure if this is ok. For example, CNaux1 = @(t)normcdf((K*(T-t))/sigma/sqrt(T-t)) )

Later, i define another function handle integrand= N(a(t)).*b(t).*C(d(t)) and i use the quad (integrand, 0, T) (T previously defined)

I thinh i probably made more than one mistake, i am quite a bit lost...

Thank you very much!

Tom Lane 2013-3-27

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This works:

K = 1; T = 1; sigma = 2; f = @(t)normcdf((K*(T-t))/sigma./sqrt(T-t));
quad(f,0,1)

If you tried something else that does not work, you should say what you tried.

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Answer 1

Tom Lane 2013-3-27

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You didn't really explain what you tried. Here's an example that works:

quad(@(x) x.*normcdf(x),-1,1)

If I use matrix multiplication "*" instead of elementwise multiplication ".*" I get the same error you report. So if your integrand is something like this, maybe you need to switch to elementwise multiplication.

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Javier 2013-3-27

Thanks, i did not know that. However, i still can´t solve the integral (now the error message is Undefined function 'times' for input arguments of type 'function_handle)

What i have to do is to compute a definite integral for the function:

N(a(t))*b(t)*C(d(t)) dt

Where N(a(t)) is a normal probability function valued at the value a, which is in fact a function that depends on t. C(d(t)) is a cumulative normal probability function valued d, which is also a function that depends on t. And b(t) is another function of t.

I have tried to define first the functions a(t) b(t) and c(t) as function handles. Then i used the commandsnormcdf and normpdf to compute the normal cumulative and probability function on this function (i think this way i obtain also a function handle that depends on t, but i am not sure if this is ok. For example, CNaux1 = @(t)normcdf((K*(T-t))/sigma/sqrt(T-t)) )

Later, i define another function handle integrand= N(a(t)).*b(t).*C(d(t)) and i use the quad (integrand, 0, T) (T previously defined)

I thinh i probably made more than one mistake, i am quite a bit lost...

Thank you very much!

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Answer 2

Youssef Khmou 2013-3-27

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编辑：Youssef Khmou 2013-3-27

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hi,

I am no sure about the error you get, but try this version :

 m=0;
 s=1;
 x=0:0.1:10;
 C=normcdf(x,m,s);
 PHI=2*exp(j*2*pi*x); % Consider it as wave  function
 N=abs(PHI).^2; % consider it as the probability of existence .
 b=exp(-(x-2).^2);; % the thrid function as you consider
 II=C.*b.*N;
 figure, plot(x,II);
 F=trapz(x,II);