Chapter 4: Problem 4
Let \(X_{1}, X_{2}, X_{3}, X_{4}\) be four iid random variables having the same
pdf \(f(x)=\) \(2 x, 0
Chapter 4: Problem 4
Let \(X_{1}, X_{2}, X_{3}, X_{4}\) be four iid random variables having the same
pdf \(f(x)=\) \(2 x, 0
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Get started for freeLet the random variable \(Y_{n}\) have a distribution that is \(b(n, p)\). (a) Prove that \(Y_{n} / n\) converges in probability \(p .\) This result is one form of the weak law of large numbers. (b) Prove that \(1-Y_{n} / n\) converges in probability to \(1-p\). (c) Prove that \(\left(Y_{n} / n\right)\left(1-Y_{n} / n\right)\) converges in probability to \(p(1-p)\).
Let \(S^{2}\) be the sample variance of a random sample from a distribution with variance \(\sigma^{2}>0\). Since \(E\left(S^{2}\right)=\sigma^{2}\), why isn't \(E(S)=\sigma ?\) Hint: Use Jensen's inequality to show that \(E(S)<\sigma\).
Let \(Y_{n}\) denote the maximum of a random sample from a distribution of the continuous type that has cdf \(F(x)\) and pdf \(f(x)=F^{\prime}(x)\). Find the limiting distribution of \(Z_{n}=n\left[1-F\left(Y_{n}\right)\right]\)
Let \(X_{n}\) and \(Y_{n}\) be \(p\) dimensional random vectors such that \(X_{n}\) and \(\mathbf{Y}_{n}\) are independent for each \(n\) and their mgfs exist. Show that if $$ \mathbf{X}_{n} \stackrel{D}{\rightarrow} \mathbf{X} \text { and } \mathbf{Y}_{n} \stackrel{D}{\rightarrow} \mathbf{Y} $$
Let the random variable \(Z_{n}\) have a Poisson distribution with parameter \(\mu=n\). Show that the limiting distribution of the random variable \(Y_{n}=\left(Z_{n}-n\right) / \sqrt{n}\) is normal with mean zero and variance \(1 .\)
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