Question: Please solve (i) and (ii)(b) only Al. (i) Let X be a random variable with E[|X]] 0. Prove the Chebyshev inequality: for any & >

Please solve (i) and (ii)(b) only

Al. (i) Let X be a random variable with E[|X]] 0. Prove the Chebyshev inequality: for any & > 0, P ( IX | ZE )S E[IX | ~ ] Ea [5 marks] (ii) Let Xn, n 2 1, be a sequence of random variables with P(Xn = Vn+1) = 1 n2 ' P(Xn = n2 (a) Prove that E[IXn - 12] - 0 as n -+ 0o. [3 marks] (b) Use the Chebyshev inequality to deduce that for any E > 0, lim P(IXn - 1| 2 8) = 0

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