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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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Show, for \(k=1,2, \ldots, n\), that $$\int_{p}^{1} \frac{n !}{(k-1) !(n-k) !} z^{k-1}(1-z)^{n-k} d z=\sum_{x=0}^{k-1}\left(\begin{array}{l}n \\\x\end{array}\right) p^{x}(1-p)^{n-x} .$$ This demonstrates the relationship between the cdfs of the \(\beta\) and binomial distributions.
Let \(X_{1}, X_{2}, X_{3}\) be iid random variables each having a standard normal distribution. Let the random variables \(Y_{1}, Y_{2}, Y_{3}\) be defined by $$X_{1}=Y_{1} \cos Y_{2} \sin Y_{3}, \quad X_{2}=Y_{1} \sin Y_{2} \sin Y_{3}, \quad X_{3}=Y_{1} \cos Y_{3}$$ where \(0 \leq Y_{1}<\infty, 0 \leq Y_{2}<2 \pi, 0 \leq Y_{3} \leq \pi\). Show that \(Y_{1}, Y_{2}, Y_{3}\) are mutually independent.
Compute \(P\left(X_{1}+2 X_{2}-2 X_{3}>7\right)\) if \(X_{1}, X_{2}, X_{3}\) are iid with common distribution \(N(1,4)\).
Let \(X\) have a gamma distribution with pdf
$$f(x)=\frac{1}{\beta^{2}} x e^{-x / \beta}, \quad 0
Show that the graph of a pdf \(N\left(\mu, \sigma^{2}\right)\) has points of inflection at \(x=\mu-\sigma\) and \(x=\mu+\sigma\).
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