# Tricky problem in statistics... Help please... 10 Point reward!?

Can someone help me with this one?

We start by dividing the interval [0,1] in k equal subintervals and then generate n independent Re(0,1)-distributed random numbers. Let now Xi be the number of random numbers that goes into subinterval i.

What is the distribution of X1, if X1 is the number of random numbers that would end up in subinterval 1? (This distribution should depend on k and n).

Now if we determine the percentage of the observations that fall into the first subinterval, Y1 = X1/n. What is the expected value, variance and standard deviation of the Y1?

Can someone please show me ALL the steps? I am quite puzzled by this problem and don't know how to approach it. I would appreciate if you outlined the entire solution and not just hints. Thanx!

Relevance

Each individual random number has a 1/k chance of falling in subinterval 1. So the distribution of X1 follows a binomial distribution:

p(X1 = i) = C(n, i) * (1/k)^i * (1-1/k)^(n-i)

Where C(n,i) = n! / (i! * (n-i)!), and * denotes multiplication.

A binomial distribution has mean np and variance np(1-p). In this case p = 1/k, so:

mean(X1) = n/k

variance(X1) = n/k * (1 - 1/k) = n/k * (k-1)/k

If we scale by 1/n (from X1 to Y1), then the mean scales by 1/n and the variance scales by 1/n^2, so:

mean(Y1) = 1/k

variance(Y1) = 1/nk * (k-1)/k

(And standard deviation is square root of variance).