Chapter 1: Problem 4
Given \(\int_{C}\left[1 / \pi\left(1+x^{2}\right)\right] d x\), where \(C \subset
\mathcal{C}=\\{x:-\infty
Chapter 1: Problem 4
Given \(\int_{C}\left[1 / \pi\left(1+x^{2}\right)\right] d x\), where \(C \subset
\mathcal{C}=\\{x:-\infty
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Get started for freeLet \(X\) denote a random variable for which \(E\left[(X-a)^{2}\right]\) exists. Give an example of a distribution of a discrete type such that this expectation is zero. Such a distribution is called a degenerate distribution.
Find the pdf \(f(x)\), the 25 th percentile, and the 60 th percentile for each
of the following cdfs: Sketch the graphs of \(f(x)\) and \(F(x)\).
(a) \(F(x)=\left(1+e^{-x}\right)^{-1},-\infty
If the variance of the random variable \(X\) exists, show that $$E\left(X^{2}\right) \geq[E(X)]^{2}$$
Let \(f(x)=2 x, 0
Let \(X\) be a random variable such that \(P(X \leq 0)=0\) and let \(\mu=E(X)\) exist. Show that \(P(X \geq 2 \mu) \leq \frac{1}{2}\).
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