I am trying to find the cumulative distribution function of the the random variable (X), where X is defined as the following:
X = Sum ( B_k * k^-2 , {k,1,Infinity} ), where B_k is a vector of i.i.d Bernoulli random variables with common success probability (p). It is given that (k) is positive real.
Any guesses on how to obtain an analytical closed form for the CDF of (X) ?
Dear Akram,
Look at the following article
http://www.math.uha.fr/chevallier/publication/binomiale5.pdf
titled “Law of the sum of Bernoulli random variables”. Maybe thanks to it you will find your solution..
Thanks Omid, I have tried both the characteristic function and the Laplace transform, you can reach a certain point, but however, the inversion to CDF does not seem to be tractable.
Thanks Radoslaw for suggesting this document. My problem is similar to the sum of Bernoulli random variables, but I am trying to find the sum of "scaled" Bernoulli random variables, where the scale decays exponentially k^-2.
We can also see this problem as an extension of Euler–Riemann zeta function having some statistical distribution depending on the Bernoulli parameter (p)
Dear Akram,
Did you think about doing numerical experiment with your problem? You can try to use the appropriate mathematical/statistical software and see the result. Of course it is the first step to image your analytical result. I can try to help you with Mathematica software. This program can generate random variables (for examples Bernoulli random variables) and possess CDF command. I do not guarantee the result but if you write me precisely what to do we can try to do such an experiment.
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