Math Solutions

[Solution] (a) Let X_1, X_2, ... be a sequence of random variables which converges to μ in probability. Let Y_n=a X_n+b for some constants a and b with

Question: (a) Let \(X_{1}, X_{2}, \ldots\) be a sequence of random variables which converges to \(\mu\) in probability. Let \(Y_{n}=a X_{n}+b\) for some constants \(a\) and \(b\) with \(a>0\). Show that \(Y_{n}\) converges to \(a \mu+b\) in probability.

(b) If \(X_{n} \sim \operatorname{Normal}(0, n)\), show that \(Y_{n}=X_{n} / n\) converges to 0 in probability.

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