Let $X_i$, $i\geq 1$, be independent and identically distributed random variables having the uniform distribution over $(0,1)$. Let $X$ be defined as $X=\sum_{i=1}^{N}X_i$, where $N$ is an unknown integer.
(a) Find an unbiased estimator $T(X)$ of $N$.
(b) Decide with adequate reasons, if $\dfrac{T(X)}{N}$ converges to $1$ almost surely, as $N$ goes to infinity.
Now, in my previous question I wanted to know something different (link) but not the solution. So, I tried to find the solution after @Ben's hint.
Now, we know that $E(2X)=N$. But since, $2X$ is not always natural number, I need to find the value, $E([2X]+1)$. Here, $[\cdot]$ is box function.
To do that, we need to find the distribution of $[2X]$.
Now, $P([2X]=k)=P(k\leq 2X<k+1)$, $k=0,1,\dots,2N-1$
Although, I could not find a way to compute $P(k\leq 2X<k+1)$. Any help appreciated.
You're making this too complicated. If you want the estimator to always yield integer values, just round it to the nearest integer. The invariance under $X_i\to1-X_i$ ensures that the rounding errors are unbiased.