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Chapter 5: Problem 3
Let \(Y\) be \(b\left(72, \frac{1}{3}\right)\). Approximate \(P(22 \leq Y \leq 28)\).
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Let \(\bar{X}_{n}\) denote the mean of a random sample of size \(n\) from a Poisson distribution with parameter \(\mu=1\). (a) Show that the \(\mathrm{mgf}\) of \(Y_{n}=\sqrt{n}\left(\bar{X}_{n}-\mu\right) / \sigma=\sqrt{n}\left(\bar{X}_{n}-1\right)\) is given by \(\exp \left[-t \sqrt{n}+n\left(e^{t / \sqrt{n}}-1\right)\right]\) (b) Investigate the limiting distribution of \(Y_{n}\) as \(n \rightarrow \infty\).
Let the pmf of \(Y_{n}\) be \(p_{n}(y)=1, y=n\), zero elsewhere. Show that \(Y_{n}\) does not have a limiting distribution. (In this case, the probability has "escaped" to infinity.)
Let \(\mathbf{X}_{n}\) and \(\mathbf{Y}_{n}\) be \(p\) -dimensional random vectors. Show that if $$\mathbf{X}_{n}-\mathbf{Y}_{n} \stackrel{P}{\rightarrow} \mathbf{0} \text { and } \mathbf{X}_{n} \stackrel{D}{\rightarrow} \mathbf{X}$$ where \(\mathbf{X}\) is a \(p\) -dimensional random vector, then \(\mathbf{Y}_{n} \stackrel{D}{\rightarrow} \mathbf{X}\).
Let \(\left\\{a_{n}\right\\}\) be a sequence of real numbers. Hence, we can also say that \(\left\\{a_{n}\right\\}\) is a sequence of constant (degenerate) random variables. Let \(a\) be a real number. Show that \(a_{n} \rightarrow a\) is equivalent to \(a_{n} \stackrel{P}{\rightarrow} a\)
If \(Y\) is \(b\left(100, \frac{1}{2}\right)\), approximate the value of \(P(Y=50)\).
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