convergence in probability and mean square

Question

If we have a sequence of independent random variables defined as: Y$_n$= $a$+$n$ with probability 1$/n$; and Y$_n$=$a$ with probability 1-1$/n$;

1-Is the sequence converges in probability?

2-Is the sequence converges in mean square?

if it is convergent in either sense what's the limit?

Answer 1

Hint: let $X_n=Y_n-a$. Then $P(|X_n|\geq 1)=\frac 1n$, so what can you deduce about convergence in probability?

We have $E[X_n^2]=(a+n)^2/n+a^2(1-1/n)$, so $E[X_n^2]$ is not bounded.