Chapter 1: Problem 7
Let \(f(x)=1 / x^{2}, 1
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Chapter 1: Problem 7
Let \(f(x)=1 / x^{2}, 1
These are the key concepts you need to understand to accurately answer the question.
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Get started for freeLet \(X\) be a random variable such that \(R(t)=E\left(e^{t(X-b)}\right)\) exists
for \(t\) such that \(-h
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 skeuness. 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
A French nobleman, Chevalier de Méré, had asked a famous mathematician, Pascal, to explain why the following two probabilities were different (the difference had been noted from playing the game many times): (1) at least one six in 4 independent casts of a six-sided die; (2) at least a pair of sixes in 24 independent casts of a pair of dice. From proportions it seemed to de Méré that the probabilities should be the same. Compute the probabilities of \((1)\) and \((2)\).
The random variable \(X\) is said to be stochastically larger than the random variable \(Y\) if $$P(X>z) \geq P(Y>z)$$ for all real \(z\), with strict inequality holding for at least one \(z\) value. Show that this requires that the cdfs enjoy the following property $$F_{X}(z) \leq F_{Y}(z)$$ for all real \(z\), with strict inequality holding for at least one \(z\) value.
For each of the following pdfs of \(X\), find \(P(|X|<1)\) and
\(P\left(X^{2}<9\right)\).
(a) \(f(x)=x^{2} / 18,-3
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