Chapter 4: Problem 8
Let \(Y\) be \(b(n, 0.55)\). Find the smallest value of \(n\) which is such that (approximately) \(P\left(Y / n>\frac{1}{2}\right) \geq 0.95\).
Chapter 4: Problem 8
Let \(Y\) be \(b(n, 0.55)\). Find the smallest value of \(n\) which is such that (approximately) \(P\left(Y / n>\frac{1}{2}\right) \geq 0.95\).
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Get started for freeLet \(X\) 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 \(Y\) denote the sum of the observations of a random sample of size 12 from
a distribution having pmf \(p(x)=\frac{1}{6}, x=1,2,3,4,5,6\), zero elsewhere.
Compute an approximate value of \(P(36 \leq Y \leq 48)\). Hint: Since the event
of interest is \(Y=36,37, \ldots, 48\), rewrite the probability as
\(P(35.5
Let \(X_{1}\) and \(X_{2}\) have a joint distribution with parameters \(\mu_{1}, \mu_{2}, \sigma_{1}^{2}, \sigma_{2}^{2}\), and \(\rho\). Find the correlation coefficient of the linear functions of \(Y=a_{1} X_{1}+a_{2} X_{2}\) and \(Z=b_{1} X_{1}+b_{2} X_{2}\) in terms of the real constants \(a_{1}, a_{2}, b_{1}, b_{2}\), and the parameters of the distribution.
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 \(X\) and \(Y\) have the parameters \(\mu_{1}, \mu_{2}, \sigma_{1}^{2}, \sigma_{2}^{2}\), and \(\rho .\) Show that the correlation coefficient of \(X\) and \(\left[Y-\rho\left(\sigma_{2} / \sigma_{1}\right) X\right]\) is zero.
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