Hi, I'm really stuck on this last question for one of my stats papers, can anyone help on it please? Thanks:
"(i) Let X be a random variable on a discrete probability space (Ω, A, P ) and let \(\displaystyle \epsilon>\epsilon_0\). Show that \(\displaystyle P(|X-a|\geq\epsilon)\leq P(|X-a|\geq\epsilon_0)\) for real and positive \(\displaystyle a\). Why is this useful when checking on stochastic convergence?
(ii) Let \(\displaystyle X_n\ for\ n\in N_0\), be a collection of random variables such that each \(\displaystyle X_n\) takes values in the set {3, 8, 11} with the distribution of \(\displaystyle X_n\) given by
\(\displaystyle p_{X_n}(3)=1-\frac{3}{n}-\frac{5}{n^2},\ \ \ P_{X_n}(8)=\frac{3}{n},\ \ \ P_{X_n}(11)=\frac{5}{n^2}\)
Prove that \(\displaystyle X_n\)→ 3 stochastically as n → ∞."
"(i) Let X be a random variable on a discrete probability space (Ω, A, P ) and let \(\displaystyle \epsilon>\epsilon_0\). Show that \(\displaystyle P(|X-a|\geq\epsilon)\leq P(|X-a|\geq\epsilon_0)\) for real and positive \(\displaystyle a\). Why is this useful when checking on stochastic convergence?
(ii) Let \(\displaystyle X_n\ for\ n\in N_0\), be a collection of random variables such that each \(\displaystyle X_n\) takes values in the set {3, 8, 11} with the distribution of \(\displaystyle X_n\) given by
\(\displaystyle p_{X_n}(3)=1-\frac{3}{n}-\frac{5}{n^2},\ \ \ P_{X_n}(8)=\frac{3}{n},\ \ \ P_{X_n}(11)=\frac{5}{n^2}\)
Prove that \(\displaystyle X_n\)→ 3 stochastically as n → ∞."