- #1

- 2

- 0

## Homework Statement

## Homework Equations

[/B]

Definition: A sequence [itex] X_1,X_2,\dots [/itex] of real-valued random variables is said to

**converge in distribution**to a random variable [itex]X[/itex] if [itex]\lim_{n\rightarrow \infty}F_{n}(x)=F(x)[/itex] for all [itex]x\in\mathbb{R}[/itex] at which [itex]F[/itex] is continuous. Here [itex]F_n, F[/itex] are the cumulative distributions functions of the random variables [itex]X_n[/itex] and [itex]X[/itex] respectively.

## The Attempt at a Solution

I'm trying to understand/recreate the following solution to the problem.

**
**

My working so far is that

$$F_{X}(x)=P(X\leq x)=\begin{cases} 0, &x<-1 \\ 1/2, &x\in[-1,1) \\ 1, &x\geq 1\end{cases}$$ and since [itex]X[/itex] only takes values 1 and -1 then [itex]X_n = (-1)^{n+X}+\frac1n=(-1)^{n+1}+\frac1n[/itex] and so $$F_{X_n}(x)=P(X_n\leq x)=\begin{cases} 0, &x<(-1)^{n+1}+\frac{1}{n} \\ 1, &x\geq (-1)^{n+1}+\frac{1}{n}\end{cases}$$ I can't understand how the limits to this have been achieved in the solution. Why does [itex]F_{X_n}(x)\rightarrow 1/2[/itex] for [itex]t\in(-1,1)[/itex], say?

Last edited: