If $U$ is a uniform random variable on $[0,1]$, what is the distribution of the random variable $X=[nU]$, where $[t]$ denotes the greatest integer less than or equal to $t$?
I've tried searching around for the solution but to no avail. I'm not really sure how to do this question and any help would be much appreciated.
On
First note that $nU\sim U(0,n)$ and the probability that this intermediate random variable falls within a bin of length 1 is $\frac1n$. Taking floors discretises the variable into such bins; there are $\lfloor n\rfloor$ length-1 bins and one last bin of length $n-\lfloor n\rfloor$. Thus $\lfloor nU\rfloor$ is a discrete random variable supported on $[0,\lfloor n\rfloor]$ with distribution $$P(\lfloor nU\rfloor=x)=\begin{cases}\frac1n&0\le x<\lfloor n\rfloor\\\frac{n-\lfloor n\rfloor}n&x=\lfloor n\rfloor\end{cases}$$ In particular, if $n$ is a natural number this reduces to a discrete $U(0,n-1)$ distribution (both bounds inclusive).
Let $n=2.$ Then $[2U]=0$ when $0 \le U < 1/2$ and $[2U]=1$ when $1/2\le U < 1.$
($[2U]=2$ for $U=1,$ but $P(U=1)=0,$ so this case doesn't matter.)
So, since $U$ has an equal chance of being in $[0,1/2)$ as $(1/2,1),$ $[2U]$ has a $1/2$ probability of being zero and a $1/2$ probability of being one.
Now try it for $n=3$ and generalize.