Chapter 4: Problem 5
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.)
Chapter 4: Problem 5
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.)
All the tools & learning materials you need for study success - in one app.
Get started for freeFind the variance of the sum of 10 random variables if each has variance 5 and if each pair has correlation coefficient \(0.5 .\)
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 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 \(n \rightarrow \infty\). Hint: Replace, by its MacLaurin's series, the expression \(e^{t / \sqrt{n}}\), which is in the exponent of the mgf of \(Y_{n}\).
Let \(W_{n}\) denote a random variable with mean \(\mu\) and variance \(b / n^{p}\), where \(p>0, \mu\), and \(b\) are constants (not functions of \(n\) ). Prove that \(W_{n}\) converges in probability to \(\mu\). Hint: Use Chebyshev's inequality.
Let \(X\) be \(N\left(\mu, \sigma^{2}\right)\) and consider the transformation \(X=\log (Y)\) or, equivalently, \(Y=e^{X}\) (a) Find the mean and the variance of \(Y\) by first determining \(E\left(e^{X}\right)\) and \(E\left[\left(e^{X}\right)^{2}\right]\), by using the mgf of \(X\). (b) Find the pdf of \(Y\). This is the pdf of the lognormal distribution.
. Let \(X\) and \(Y\) be random variables with means \(\mu_{1}, \mu_{2} ;\) variances \(\sigma_{1}^{2}, \sigma_{2}^{2}\); and correlation coefficient \(\rho .\) Show that the correlation coefficient of \(W=a X+b, a>0\), and \(Z=c Y+d, c>0\), is \(\rho\)
What do you think about this solution?
We value your feedback to improve our textbook solutions.