Nice fact! How about this for a proof: Let X_j be a sequence for unif(0,1) random variables, S_n the sequence of partial sums (so S_4=X_1+X_2+X_3+X_4).
If N is the number of uniforms needed to have a sum bigger than 1, you can see that P(N>n)=P(S_n<1).
P(S_n<1) is not too bad to compute, or you can wikipedia it (Irwin-Hall distribution) to find the expression P(S_n<1)=1/n!
Combining the things, E[N]=sum(P(N>n),n=0..infinity)=sum(1/n!,n=0..infinity)=exp(1).
Thank you, I will look into this. I've taken discrete math and some graph theory, but never stats, so all those families of distributions are alien to me :)
If N is the number of uniforms needed to have a sum bigger than 1, you can see that P(N>n)=P(S_n<1).
P(S_n<1) is not too bad to compute, or you can wikipedia it (Irwin-Hall distribution) to find the expression P(S_n<1)=1/n!
Combining the things, E[N]=sum(P(N>n),n=0..infinity)=sum(1/n!,n=0..infinity)=exp(1).