MathDB

We throw

N

balls into

n

urns, one by one, independently and uniformly. Let X_i\equal{}X_i(N,n) be the total number of balls in the

i

th urn. Consider the random variable y(N,n)\equal{}\min_{1 \leq i \leq n}|X_i\minus{}\frac Nn|. Verify the following three statements: (a) If

n \rightarrow \infty

and

N/n^3 \rightarrow \infty

, then P \left(\frac{y(N,n)}{\frac 1n \sqrt{\frac Nn}}0 \ . (b) If

n\rightarrow \infty

and

N/n^3 \leq K

(

K

constant), then for any

\varepsilon > 0

there is an

A > 0

such that P(y(N,n) < A) > 1\minus{}\varepsilon . (c) If

n \rightarrow \infty

and

N/n^3 \rightarrow 0

then

P(y(N,n) < 1) \rightarrow 1.

P. Revesz

Problem Statement