I am trying to simulate n random discrete variable which has the following pmf
$P(X = k) = (1-p)^2kp^{k-1}$
I am thinking about using the inverse transform sampling method and I am trying to find the cdf.
$P(X \le k) = 1 - P(X > k) = 1- P(X \ge k+1) = \sum_{x=k+1}^{\infty} (1-p)^2xp^{x-1} = 1-[ (1-p)^2\sum_{x=k+1}^{\infty}xp^{x-1}]$
= $1-(1-p)^2[(k+1)p^k + (k+2)p^{k+1}+ (k+3)p^{k+2} + ...] $
What I have done so far :
Let $S = (k+1)p^k + (k+2)p^{k+1}+ (k+3)p^{k+2} + ...$
$pS =$ $(k+1)p^{k+1} + (k+2)p^{k+2}+ (k+3)p^{k+3} + ...$
then $(1-p)S = kp^k +p^k + p^{k+1} + p^{k+2}+p^{k+3}+...$ (geometric series )
$(1-p)S = kp^k + \frac{p^k}{1-p}$
$ F(x) = 1- (xp^x(1-p) + p^x)$
Then I'm trying to find the inverse :
$1- (xp^x(1-p) + p^x) < U \le 1- ((x+1)p^{x+1}(1-p) + p^{x+1})$
$xp^x(1-p) + p^x < 1-U < (x+1)p^{x+1}(1-p) + p^{x+1}$
I'm stuck here ...
Any help or hint will be appreciated !
There are several ways you can randomly sample from a discrete distribution, unfortunately inverting the CDF is not one of them. The plot below was generated using the Alias Method, it is particularly efficient if you know how to use binary search trees, otherwise a simple implementation of
argmin,argmaxwill workThe blue bars are a histogram of $400$ samples generated with the alias method, the red points are simply
$$ P(X = x) = (1 - p)^2 x p^{x - 1} $$