Let $X_1, \dots, X_c$ be independent uniform random variables on $[0,1]$. We know the number of random variables whose value will fall in the range $[0,1]$ will always be exactly $c$. Now define $Z_i = X_i + Y_i$ where $Y_i$ are independent random variables that are exponentially distribution. Let $C_Z$ be the number of random variables $Z_i$ whose value falls in the range $[0,1]$. I am interested in the distribution of $C_Z$.
If we set $c=100$ and $\lambda = 1$ then, by simulation the pdf of $C_Z$ looks like:
I fitted a normal distribution to the data and have drawn that on top of the histogram created by simulation. This is just to show the data appears to be approximately normal.
It seems that $C_Z$ has a shifted binomial distribution. Is that right and if so, why?
@E-A points out that the pdf is binomial with parameters (p, c) with $p = P(X_i + Y_i \leq 1)$. But what is $P(X_i + Y_i \leq 1)$? By simulation again I plotted $P(X_i + Y_i \leq 1)$ for $0 \leq \lambda \leq 5$. Does this have a simple form?