Determine $P(S_n\leq1)$ where $S_n=\sum_{k=1}^nX_k$

Question

Suppose that $X_n$ are i.i.d. $Uniform(0,1)$ random variables. Let $S_n=\sum_{k=1}^nX_k$ with $S_0:=0$. Then, determine $P(S_n\leq1)$.

I know that maybe by using Characteristic function of $S_n$ I will be able to get the d.f. of $S_n$ using Inversion Theorem but I do not want to use that.

I tried the following method:

I want to find $P(S_n>1)$ instead. Let $A=\{S_n>1\}$. Then define $A_k=\{S_{k-1}\leq1,S_k>1\}$. Notice that $A_k$ are disjoint, and $\cup_kA_k=A$.

Hence $P(A)=\sum_{k=1}^nP(A_k)=\sum_{k=1}^nP(S_{k-1}\leq1,S_k>1)=\sum_{k=2}^nP(S_{k-1}\leq1,S_k>1)$ because $P(S_0\leq1,S_1>1)=0$.

Now $P(S_{k-1}\leq1,S_k>1)=P(S_{k-1}\leq1, S_{k-1}>1-X_k)=P(1-X_k< S_{k-1}\leq 1)$

It seems I can't proceed from here, because I would need knowledge of $P(S_{k-1}\leq1)$. Will induction work here, then?

In general is there any formula for $P(S_n\leq a)$ where $a>0$?

Answer 1

Note that the vector $(X_1,X_2,...,X_n)$ has density $f(x_1,x_2,...,x_n)=1$ on the unit hypercube. So $\mathbf{P}(S_n\leq 1)$ is just the volume of the region bounded by the axis planes and the plane $x_1+x_2+\cdots+x_n=1$ and equals $1/{n!}$.