MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Suppose we have a family $F_0,F_1,\dots$ of independent random variables which take the value $1$ with probability $p$ and $0$ otherwise; let $\delta$ be a number between $0$ and $1$. Let

$X_n = \sum_{k=0}^n \delta^{n-k} F_k$.

I'm interested in the distribution of $X_n$. It seems straightforward enough to be known and have a name - does anybody know what it is?

share|cite|improve this question
In the formula you probably meant $F_k$ instead of $F_n$. Also, replacing $n-k$ with $k$ seems more natural. – Ori Gurel-Gurevich Oct 11 '10 at 17:41
You are right about the $k$. Using $n-k$ rather than $k$ is more natural in the context I'm considering but of course it makes no practical difference for individual $n$. – Tom Smith Oct 12 '10 at 5:46
I only mentioned that because then you can take $n=\infty$. – Ori Gurel-Gurevich Oct 13 '10 at 5:09

Unless I misunderstood your intention (see my comment above), if you take $n=\infty$ you get a Bernoulli convolution. See the paper Sixty Years Of Bernoulli Convolutions by Peres, Schlag and Solomyak which can also the last paper here.

share|cite|improve this answer
The URL above doesn't work for some reason. – Peter Shor Oct 12 '10 at 1:49
I also get an error. I added a link to the paper on Boris Solomyak's homepage. – Ori Gurel-Gurevich Oct 12 '10 at 3:53
Thanks - I'll take a look at that later when I have time. – Tom Smith Oct 12 '10 at 5:47

It is a finite distribution over numbers which aren't necessarily integers. I wouldn't expect it to have a name.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.