Let $Y_1$, $Y_2$, ... be a sequence of random variables such that $P(Y_n=\frac{1}{n})=1-\frac{1}{n^2}$ and $P(Y_n=n)=\frac{1}{n^2}$. Does $Y_n$ converge in probability?
I am stuck because I don't know how to deal with RV with two outcomes. And I am confused with the different probability in which $Y_n=1/n$ and $Y_n=n$. I know how to prove convergence with a single $Y_n$ to Y. But really stuck in this one.
so $$ Y_1=\begin{cases} 1 &\mbox{p } = 0 \\ 1 &\mbox{p } = 1 \end{cases} $$ $$ Y_2=\begin{cases} 2 &\mbox{p } = \frac{1}{4} \\ \frac{1}{2} &\mbox{p } = \frac{3}{4} \end{cases} $$
and then I don't know how to prove whether $Y_n$ converges or not in probability.
any help will be great. thanks