Trying to solve the following question. Not sure if it is a trivial question or I am missing something here:
I have a random variable $X$, with:
$P(X=1) = P(X=-1) = 1/2$
Now define $X_n$ as:
$$ X_n= \begin{cases} X,& \text{with probability}\ 1-1/n\\ e^n, & \text{with probability}\ 1/n \end{cases} $$
Does $X_n$ converge to $X$ in probability and distribution? My intuition says yes, as $n$ approaches infinity, $1/n$ will approach $0$ and $X_n=X$. Is this a valid proof?
Thanks!