How Do You Solve Complex Fourier Integrals in Probability Theory?

[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

 

[suyver]

$$f(q)=ab\int_{-\infty}^{\infty}x^{b-1}\exp(-a\,x^b) \exp(i \,q\,x)dx$$

I don't think that this integral has an analytical solution...

 

[tacman]

for sharing your question and the equations that you are trying to solve. I am not able to provide a specific solution to this Fourier integral without more context and information about the variables and constants involved. However, I can provide some general steps that may help you in solving this integral.

Step 1: Understand the problem and the variables involved

It is important to have a clear understanding of the problem and the variables involved before attempting to solve the integral. In this case, it seems like the variable x represents a random variable and q is the frequency variable in the Fourier transform. The variables a and b are constants that are part of the function f(x) that you are trying to integrate.

Step 2: Use known Fourier transform pairs

If possible, try to use known Fourier transform pairs to simplify the integral. This can help in reducing the integral to a more manageable form. For example, if the function f(x) is a Gaussian function, you can use the Fourier transform pair for the Gaussian function to simplify the integral.

Step 3: Use integration by parts

If the integral cannot be simplified using known Fourier transform pairs, you can try using integration by parts to solve it. This technique involves breaking down the integral into smaller parts and using the integration by parts formula to solve each part. This can help in reducing the complexity of the integral and making it easier to solve.

Step 4: Apply the convolution theorem

As mentioned in the equations provided, the convolution theorem can be used to solve 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 this case, you can use this theorem to simplify the integral by taking the Fourier transform of each individual function and then multiplying them together.

Step 5: Solve the inverse Fourier transform

Once you have simplified the integral using the steps above, you will be left with the inverse Fourier transform of the function f(q). This can be solved using known inverse Fourier transform pairs or by using numerical methods if needed.

In conclusion, solving Fourier integrals can be a complex process and may require some knowledge of advanced mathematics. I hope the steps provided above can help guide you in solving your specific integral. It is always helpful to consult with a mathematics expert or use online resources for further assistance. Best of luck in solving your problem!