Probability of $n$ random reals summing up less than $k$

Question

There are $n$ random numbers uniformly generated from $0$ to $1$. What is the probability of its sum less than $k$? For cases like $k=1$, it is easy to use the geometry method. However, the same method is no longer trivial for other values. Is there a good way to derive a general formula?

Answer 1

Let $X_1,...,X_n$ iid where $X_i$ is unioform in $[0,1]$. If $n$ is big enough, $$\frac{X_1+...+X_n-n\mathbb E[X_1]}{\sqrt{nVar(X_1)}}$$

can be approximated by a $\mathcal N(0,1)$ law.

Answer 2

The sum of $n$ independent $U(0,1)$ is the Irwin-Hall distribution. You can find the CDF and more on the wiki page.

Essentially, you Fourier invert the $n$-th power of the very simple characteristic function of $U(0,1)$ $\varphi(t)=\frac{\exp(it)-1}{it}$ to get the probabilty density function, hence the CDF.

Answer 3

The probability that the sum lies in the interval $[k,k+1]$ where $k$ is an integer is $$\frac{A(n,k)}{n!}$$ where $A(n,k)$ is an Eulerian number. So the probability that the sum is $\le k$ for $k$ an integer is $$\frac{1}{n!}\sum_{j=0}^{k-1}A(n,j).$$