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Chapter 1: Problem 2
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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If a sequence of sets \(C_{1}, C_{2}, C_{3}, \ldots\) is such that \(C_{k} \subset C_{k+1}, k=1,2,3, \ldots\), the sequence is said to be a nondecreasing sequence. Give an example of this kind of sequence of sets.
For each of the following cdfs \(F(x)\), find the pdf \(f(x)[\mathrm{pmf}\) in
part \((\mathrm{d})]\), the 25 th percentile, and the 60 th percentile. Also,
sketch the graphs of \(f(x)\) and \(F(x)\).
(a) \(F(x)=\left(1+e^{-x}\right)^{-1},-\infty
Let \(X\) be a random variable with mean \(\mu\) and variance \(\sigma^{2}\) such
that the third moment \(E\left[(X-\mu)^{3}\right]\) about the vertical line
through \(\mu\) exists. The value of the ratio \(E\left[(X-\mu)^{3}\right] /
\sigma^{3}\) is often used as a measure of skewness. Graph each of the
following probability density functions and show that this measure is
negative, zero, and positive for these respective distributions (which are
said to be skewed to the left, not skewed, and skewed to the right,
respectively).
(a) \(f(x)=(x+1) / 2,-1
Let \(X\) have the uniform pdf \(f_{X}(x)=\frac{1}{\pi}\), for
\(-\frac{\pi}{2}
Let the three mutually independent events \(C_{1}, C_{2}\), and \(C_{3}\) be such that \(P\left(C_{1}\right)=P\left(C_{2}\right)=P\left(C_{3}\right)=\frac{1}{4} .\) Find \(P\left[\left(C_{1}^{c} \cap C_{2}^{c}\right) \cup C_{3}\right]\)
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