# What is this probability distribution?

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?

• 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.