Solving Fourier Integral for Random Variable Sum

[capitano_nemo]

Hi to everybody

I have to solve this Fourier integral:

1) f(q)=\int_{-infty}^{+infty}a*b*x^(b-1)*exp(-a*x^b)*exp(i*q*x)*dx

and if S_n=x_1+...+x_n, with S_n the sum of n random variables IID, then I can write:

f_n(q)=[f(q)]^n,(convolution theorem), then the anti-trasform of f_n(q) give the pdf of the variable S_n.

2) F(S_n)=(1/2*pi)*\int_{-infty}^{+infty}f_n(q)*exp(-i*q*x)*dq.

I must to solve the equations 1) and 2) in order to solve my problem, the equation 2) is the final solution of the problem.

Thanks

 

[member 553083]

in advance!</code>Thank you for your help. I think I got it now. The solution is as follows:f(q) = a*b*∫-∞ to +∞ x^(b-1)*exp(-a*x^b)*exp(i*q*x)dxf_n(q) = [f(q)]^n (convolution theorem)F(S_n) = (1/2π)*∫-∞ to +∞ f_n(q)*exp(-i*q*x)dqSo, the solution is F(S_n) = (1/2π)*∫-∞ to +∞ [a*b*∫-∞ to +∞ x^(b-1)*exp(-a*x^b)*exp(i*q*x)dx]^n *exp(-i*q*x)dqHope this helps!

 

[M98Ranger]

for sharing your question with us. The Fourier integral that you have provided is quite complex and it can be challenging to solve. However, there are a few steps that you can follow to solve it.

Firstly, you can start by using the convolution theorem to simplify the integral. This theorem states that the Fourier transform of the convolution of two functions is equal to the product of their individual Fourier transforms. In your case, this means that you can write f_n(q) as the product of n copies of f(q).

Next, you can use the properties of the Fourier transform to simplify the integral further. For example, the Fourier transform of a product of two functions is equal to the convolution of their individual Fourier transforms. Additionally, the Fourier transform of the exponential function is a delta function. These properties can help you simplify the integral and make it easier to solve.

Once you have simplified the integral, you can use the inverse Fourier transform to get the pdf of the random variable S_n. This involves integrating over all values of q and then taking the inverse Fourier transform of the result.

In order to solve the equations, you may need to use numerical methods or software such as MATLAB or Mathematica. These tools can help you evaluate the integrals and solve the equations to get the final solution.

I hope this helps you in solving your problem. Best of luck!