Can someone help me with the following question:
For each $n\geq 1$, let $X_{n}$ be a random variable following an exponential distribution with mean $n$. Determine $F_{n}$: the distribution function of $X_{n} - 10 [\frac{X_{n}}{10}]$, where $[u]$ denotes the greatest integer lesser or equal than $u$. What is the limit of $F_{n}$ when $n \rightarrow + \infty$.
I've tried to calculate that by using Probability Total Law, but since I have the same variable both in $[X_{n}/10]$ and out the floor function, I was not able to make it.
Using computational simulation I've found that the answer should be a Uniform distribution (0, 10) but I can't prove it mathematically.
$F_n$ can be computed readily. Let $Y_n=X_n-10[\frac{X_n}{10}]$. Then for $0\le y \le 10$, $P(Y_n \le y)=\sum_{k=0}^{\infty}P(10k\le X_n\le 10k+y)=\sum_{k=0}^{\infty}(e^{\frac{-10k}{n}}-e^{\frac{-(10k+y)}{n}})=(1-e^{\frac{-y}{n}})\frac{1}{1-e^{\frac{-10}{n}}}$ For large n, $F_n(y)\approx \frac{y}{10}$, which is uniform for $0\le y\le 10$.
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