8 (10 pts). Let X, X2, be a sequence of random variables. We consider two different...

 

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8 (10 pts). Let X₁, X2, be a sequence of random variables. We consider two different conver- gences of a sequence of random variables. (1) convergence in the 2nd mean: Xn converges in the 2nd mean to a random variable X if E|Xn - X² → 0 as n →∞. (2) convergence in probability: Xn converges in probability to a random variable X if for every € > 0, P(|Xn - X| > €) → 0 as n→∞. (a) Prove that if Xn converges in the 2nd mean to X, Xn converges in probability to X. (b) Provide an example that Xn converges in probability to X but Xn does not converge in the 2nd mean to X. 8 (10 pts). Let X₁, X2, be a sequence of random variables. We consider two different conver- gences of a sequence of random variables. (1) convergence in the 2nd mean: Xn converges in the 2nd mean to a random variable X if E|Xn - X² → 0 as n →∞. (2) convergence in probability: Xn converges in probability to a random variable X if for every € > 0, P(|Xn - X| > €) → 0 as n→∞. (a) Prove that if Xn converges in the 2nd mean to X, Xn converges in probability to X. (b) Provide an example that Xn converges in probability to X but Xn does not converge in the 2nd mean to X.