Let $X_n$ have a continuous uniform distribution on $(0,n)$. Find $\lim_{n\to \infty} F_{X_n}(x)$ for $x>0$

Question

Let $X_n$ have a continuous uniform distribution on $(0,n)$. Find $\lim_{n\to \infty} F_{X_n}(x)$ for $x>0$. Is $\lim_{n\to \infty}F_{X_n}(x)$ continuous? Is it a cumulative distribution function?

So I computed $F_{X_n}(x)=\int_0^x\frac{1}{n}dt=\frac{x}{n}$

then $\lim_{n\to\infty}\frac{x}{n}=0$ for all $x$

So I get its a constant function $0$. I'm not sure how to answer the questions though. I believe it should be continuous since the constant function is continuous.

Answer 1

Your computation of $F_{X_n}(x)$ is valid for $x\leqslant n$ hence for a fixed $x$, it is valid for $n$ large enough (this will not change the correctedness of your limit).

Indeed, for all $x>0$, $F_{X_n}(x)\to F(x):=0$ and the function $F$ is continuous on $(0,+\infty)$. If $F$ was a cumulative distribution function, what should be $\lim_{x\to +\infty}F(x)$?