Need help getting started on probability question Let $f_n$ be the density function and $F_n$ the cdf of the sum $S_n$ of n independent uniform (0, 1) random variables. 
Show that on each of the n intervals ($i$ - $1$, $i$) for $i$ = $1$ to n, $f_n$ is equal to a polynomial of degree $n - 1$, and $F_n$ is equal to a polynomial of degree $n$. 
I'm having trouble understanding where this polynomial is coming from. Am I supposed to deduce its equation from the information given? Any hints on how to start the problem would be greatly appreciated. Thanks!
 A: As vonbrand comments,
the key here is to compute the probability density function of $S_n$.
(Hint: the distribution in question is known as the Irwin-Hall distribution.
Look at the special cases section of the Wikipedia page to have an idea of what is going on.)
Once you have established that the density function $f_n$ of $S_n$ satisfies the required property,
it shouldn't be too hard to prove that $F_n$ does too.
(Hint: remember the relationship between the probability density function and the cumulative distribution function when the former is continuous.)
A: I'd to this by induction.
For $n=1$ this is trivially true.
For $n=2$, use the formula for the density of the sum of two random variables. You don't need to bother with finding the exact coefficients of $f_2$, it suffices to show that $f|_{[0,1]}$ and $f|_{[1,2]}$ are indeed polynomials of degree $1$.
For $n+1$, proceed as for $n=2$, except that one of the summands now has density $f_{n-1}$ instead of a uniform density on $[0,1]$. Again, you're not interested in the precise coefficients of $f_{n+1}$, you just need to show that $f_{n+1}|_{[i-1,i]}$ is a polynomial of degree $n$. You will of course need to use the corresponding property of $f_n$.
Finally, use that $f_n = F_n'$ to prove that $F_n$ has the desired properties too.
