A Weird but Interesting Distribution
by 3ch03s, Nov 18, 2014, 5:51 AM
Let 
an iid bernoulli sequence variables with success parameter 
, and let compute the (random) sum:
then, when
tends to infinity, the variable 
converges to a weird distribution 
, presumibly without radon-nikodym derivative for 
and for 
tends to an uniform distribution in [0,1]. in both cases we have to take care about the non-numerability of R, because of , and diadic arguments, combined by density.
Let's call
to this distribution. this distribution has a several properties like:
a)
b) if
is bernoulli with parameter independent of 
then:
c) From (b) we can compute the mean and variance of, as has the same distribution as 
then
![$E[R]=\frac{E[X]+E[R]}2=\frac{p+E[R]}2$](//latex.artofproblemsolving.com/e/8/f/e8f795a8c30e79e91b6d6364dc6de312e5a10059.png)
Obtaining![$E[R]=p$](//latex.artofproblemsolving.com/1/a/1/1a128bc77de3fcea4211431af3a2b28777a8d2c3.png)
For the variance we have
![$V[R]=\frac 14 V[X+R]$](//latex.artofproblemsolving.com/1/5/7/157eb1c4a7c29546b3332389f352a4e178ac1d0f.png)
, and by independence:
![$V[R]=\frac 14 (V[X]+V[R])=\frac 14(p(1-p)+V[R])$](//latex.artofproblemsolving.com/b/7/b/b7b2fd886274be6757b35cf5e58dd8d5e9aebf59.png)
Hence![$V[R]=\frac 13p(1-p)$](//latex.artofproblemsolving.com/9/d/1/9d1246e3d92dcf485d27d9e200e3fa5571bc54ff.png)
d) Also from (b) we have an interesting property about the cdf of R:
![$P[R \leq r]=P[X+R \leq 2r]=P[R\leq 2r](1-p)+pP[R\leq 2r+1]$](//latex.artofproblemsolving.com/a/f/6/af66ceb53d2256acaa540783d996f93687579ce6.png)
With this formula, we can compute by recursion the cdf evaluated in any![$r \in [0,1]$](//latex.artofproblemsolving.com/f/a/0/fa0395e2b43ab1baeeaf66792f9c9e67a9eae154.png)
, and also, give a closed form when 
is rational.
From this, the cdf has an interesting property of self-similarity, and a fractal-like effect when we plot the cdf.
i'll be continue.....
an iid bernoulli sequence variables with success parameter 
, and let compute the (random) sum:
then, when
tends to infinity, the variable 
converges to a weird distribution 
, presumibly without radon-nikodym derivative for 
and for 
tends to an uniform distribution in [0,1]. in both cases we have to take care about the non-numerability of R, because of , and diadic arguments, combined by density.Let's call
to this distribution. this distribution has a several properties like:a)
b) if
is bernoulli with parameter independent of 
then:
c) From (b) we can compute the mean and variance of
, as has the same distribution as 
then![$E[R]=\frac{E[X]+E[R]}2=\frac{p+E[R]}2$](http://latex.artofproblemsolving.com/e/8/f/e8f795a8c30e79e91b6d6364dc6de312e5a10059.png)
Obtaining
![$E[R]=p$](http://latex.artofproblemsolving.com/1/a/1/1a128bc77de3fcea4211431af3a2b28777a8d2c3.png)
For the variance we have
![$V[R]=\frac 14 V[X+R]$](http://latex.artofproblemsolving.com/1/5/7/157eb1c4a7c29546b3332389f352a4e178ac1d0f.png)
, and by independence:![$V[R]=\frac 14 (V[X]+V[R])=\frac 14(p(1-p)+V[R])$](http://latex.artofproblemsolving.com/b/7/b/b7b2fd886274be6757b35cf5e58dd8d5e9aebf59.png)
Hence
![$V[R]=\frac 13p(1-p)$](http://latex.artofproblemsolving.com/9/d/1/9d1246e3d92dcf485d27d9e200e3fa5571bc54ff.png)
d) Also from (b) we have an interesting property about the cdf of R:
![$P[R \leq r]=P[X+R \leq 2r]=P[R\leq 2r](1-p)+pP[R\leq 2r+1]$](http://latex.artofproblemsolving.com/a/f/6/af66ceb53d2256acaa540783d996f93687579ce6.png)
With this formula, we can compute by recursion the cdf evaluated in any
![$r \in [0,1]$](http://latex.artofproblemsolving.com/f/a/0/fa0395e2b43ab1baeeaf66792f9c9e67a9eae154.png)
, and also, give a closed form when 
is rational.From this, the cdf has an interesting property of self-similarity, and a fractal-like effect when we plot the cdf.
i'll be continue.....
This post has been edited 1 time. Last edited by 3ch03s, Mar 24, 2016, 5:27 PM
Reason: 2^i
Reason: 2^i
March 2016
August 2015