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Chapter 3: Problem 9
Determine the 90 th percentile of the distribution, which is \(N(65,25)\).
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Let the random variable \(X\) have the pdf
$$f(x)=\frac{2}{\sqrt{2 \pi}} e^{-x^{2} / 2}, \quad 0
Let
$$f(x, y)=(1 / 2 \pi) \exp
\left[-\frac{1}{2}\left(x^{2}+y^{2}\right)\right]\left\\{1+x y \exp
\left[-\frac{1}{2}\left(x^{2}+y^{2}-2\right)\right]\right\\}$$
where \(-\infty
Let \(X\) have a conditional Burr distribution with fixed parameters \(\beta\) and \(\tau\), given parameter \(\alpha\). (a) If \(\alpha\) has the geometric pmf \(p(1-p)^{\alpha}, \alpha=0,1,2, \ldots\), show that the unconditional distribution of \(X\) is a Burr distribution. (b) If \(\alpha\) has the exponential pdf \(\beta^{-1} e^{-\alpha / \beta}, \alpha>0\), find the unconditional pdf of \(X\)
Let the number of chocolate chips in a certain type of cookie have a Poisson distribution. We want the probability that a cookie of this type contains at least two chocolate chips to be greater than \(0.99 .\) Find the smallest value of the mean that the distribution can take.
Let \(X\) and \(Y\) have a bivariate normal distribution with parameters \(\mu_{1}=\) \(20, \mu_{2}=40, \sigma_{1}^{2}=9, \sigma_{2}^{2}=4\), and \(\rho=0.6 .\) Find the shortest interval for which \(0.90\) is the conditional probability that \(Y\) is in the interval, given that \(X=22\).
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