The Irwin-Hall distribution: A Problem to do with a Sum of Uniform Distributions

The following question appears at http://www.spellscroll.com/questionfull/184/ where it appears to trace back to http://www.wilmott.com/

X_1, X_2,..., X_n are independent random variables, uniformly distributed on [0,1].

What is the probability that X_1 +X_2 +... +X_n < 1 ?

An attempt:

A general sum of independent identically distributed U[0,1] random variables seems to be called the Irwin-Hall distribution. It is clearly continuous and of interest at least in the general problem of finding the distribution of a sum of random variables.

We can see that a general X_k, k= 1,...,n has characteristic function (given as a Fourier transform of its probability density function),

ϕ_k (t) = E[exp (t.X_k)] = i.(exp(it) - 1) / t

We have that,

E[exp (t.(X_1 +X_2 +... +X_n))] = E[(exp (t.X_1)) ... (exp (t.X_n))] = E[exp (t.X_1)] ... E[exp (t.X_n)]

(this last pair of equalities hold for any independent X_1, ..., X_n in fact)

Whence the characteristic function of our sum,

ϕ(t) = E[exp (t.(X_1 +X_2 +... +X_n))] = [i.(exp(it) - 1) / t]^n (= ϕ_1 (t) ... ϕ_n (t))

The approach thus far is clear, invoking the characteristic function was due to it's sum to product property.

(continued...)