Question

Suppose that X, for n E N has p.m.f. P{X, = k/n} = 1/n for k = 1, . ..,n. Prove or disprove: Xn converges in distribution to U

where U has a uniform (0, 1) distribution.

Answer 1

To prove or disprove that Xn converges in distribution to U, where U has a uniform (0, 1) distribution, we need to show that the distribution of Xn approaches the distribution of U as n approaches infinity.

Step-by-step explanation:

To prove or disprove that Xn converges in distribution to U, where U has a uniform (0, 1) distribution, we need to show that the distribution of Xn approaches the distribution of U as n approaches infinity.

Given that X has a discrete uniform distribution with p.m.f. P{X = k/n} = 1/n for k = 1, . ..,n, the distribution of Xn can be represented by a histogram with n equally spaced bins.

As n approaches infinity, the width of each bin approaches zero, resulting in a continuous distribution that resembles a uniform (0, 1) distribution.

Therefore, we can conclude that Xn converges in distribution to U.