Uniform distribution and the convergence in probability.

Question

Sorry guys, I need your help. Can some one explain me how solve this exercise?

Let $Y$~Uniform$[0 ,1]$, and let $X_n=Y^n$. Prove that $X_n \to 0$ in probability.

Answer 1

We have that$$ E(X_n)=E(Y^n)=\int_0^1 y^n\,dy=\frac{1}{n+1}. $$ Then, using Markov's inequality, we have that$$ P(X_n>\epsilon)\leq \frac{E(X_n)}{\epsilon}=\frac{1}{(n+1)\epsilon}\to 0 $$ as $n\to\infty$. Hence, $X_n\to 0$ in probability.