Chapter 4: Problem 16
Let \(X_{1}\) and \(X_{2}\) be independent random variables with nonzero variances. Find the correlation coefficient of \(Y=X_{1} X_{2}\) and \(X_{1}\) in terms of the means and variances of \(X_{1}\) and \(X_{2}\).
Chapter 4: Problem 16
Let \(X_{1}\) and \(X_{2}\) be independent random variables with nonzero variances. Find the correlation coefficient of \(Y=X_{1} X_{2}\) and \(X_{1}\) in terms of the means and variances of \(X_{1}\) and \(X_{2}\).
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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 \(Y_{1}
Let \(Y_{1}
Let \(X_{1}\) and \(X_{2}\) have a trinomial distribution with parameters \(n, p_{1}, p_{2}\). (a) What is the distribution of \(Y=X_{1}+X_{2} ?\) (b) From the equality \(\sigma_{Y}^{2}=\sigma_{1}^{2}+\sigma_{2}^{2}+2 \rho \sigma_{1} \sigma_{2}\), once again determine the correlation coefficient \(\rho\) of \(X_{1}\) and \(X_{2}\)
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