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Chapter 3: Problem 11
Let \(X\) be \(b(2, p)\) and let \(Y\) be \(b(4, p)\). If \(P(X \geq 1)=\frac{5}{9}\), find \(P(Y \geq 1)\).
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Let \(X\) be \(N\left(\mu, \sigma^{2}\right)\) so that \(P(X<89)=0.90\) and \(P(X<94)=0.95\). Find \(\mu\) and \(\sigma^{2}\).
. Let \(X\) and \(Y\) have the joint \(\operatorname{pmf} p(x, y)=e^{-2} /[x !(y-x) !], y=0,1,2, \ldots ;\) \(x=0,1, \ldots, y\), zero elsewhere. (a) Find the mgf \(M\left(t_{1}, t_{2}\right)\) of this joint distribution. (b) Compute the means, the variances, and the correlation coefficient of \(X\) and \(Y\). (c) Determine the conditional mean \(E(X \mid y)\). Hint: Note that $$ \sum_{x=0}^{y}\left[\exp \left(t_{1} x\right)\right] y ! /[x !(y-x) !]=\left[1+\exp \left(t_{1}\right)\right]^{y} $$ Why?
. Let \(T\) have a \(t\) -distribution with 14 degrees of freedom. Determine \(b\)
so that \(P(-b
Using the computer, plot the cdf of \(\Gamma(5,4)\) and use it to guess the median. Confirm it with a computer command which returns the median, (In \(\mathrm{R}\) or S-PLUS, use the command qgamma \((.5\), shape \(=5\), scale \(=4)\) ).
Let \(X\) have a geometric distribution. Show that $$ P(X \geq k+j \mid X \geq k)=P(X \geq j) $$ where \(k\) and \(j\) are nonnegative integers. Note that we sometimes say in this situation that \(X\) is memoryless.
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