Evident that the cdf and the cdf the constant random variable are the same

Solved Step by Step With Explanation- Convergence Types for Xn

Questions

Check the following:

a) Xn​→P1 b) Xn​→qm1 c) Xn​→D1 

In the context of probability theory, these notations represent different types of convergence of random variables. Let's evaluate each of them without using the properties that one type of convergence implies another.

a) Xn → P 1 (Convergence in Probability to 1):

b) Xn → qm1 (Convergence in Quadratic Mean to 1):

This means that we want to check if the sequence of random variables Xn converges in quadratic mean (L2) to 1. To do this, we need to check if the expected value of the square of the difference between Xn and 1 goes to 0 as n goes to infinity.

E[Xn^2] = Var(Xn) + (E[Xn])^2 = 0 + 1^2 = 1.

So, we get:

This means that we want to check if the sequence of random variables Xn converges in distribution to a constant random variable 1. Convergence in distribution is characterized by the cumulative distribution function (CDF) of the random variables.

In this case, since Xn ~ Bernoulli(1) for all n, the CDF of Xn is:

The CDF of a constant random variable 1 is:

F_1(x) = {

Therefore, Xn → D1.

So,