Chapter 1: Problem 13
Consider the cdf \(F(x)=1-e^{-x}-x e^{-x}, 0 \leq x<\infty\), zero elsewhere. Find the pdf, the mode, and the median (by numerical methods) of this distribution.
Chapter 1: Problem 13
Consider the cdf \(F(x)=1-e^{-x}-x e^{-x}, 0 \leq x<\infty\), zero elsewhere. Find the pdf, the mode, and the median (by numerical methods) of this distribution.
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Get started for freeLet the subsets \(C_{1}=\left\\{\frac{1}{4}
Let \(X\) have the pdf \(f(x)=(x+2) / 18,-2
If \(P\left(C_{1}\right)>0\) and if \(C_{2}, C_{3}, C_{4}, \ldots\) are mutually disjoint sets, show that \(P\left(C_{2} \cup C_{3} \cup \cdots \mid C_{1}\right)=P\left(C_{2} \mid C_{1}\right)+P\left(C_{3} \mid C_{1}\right)+\cdots\)
A mode of a distribution of one random variable \(X\) is a value of \(x\) that
maximizes the pdf or pmf. For \(X\) of the continuous type, \(f(x)\) must be
continuous. If there is only one such \(x\), it is called the mode of the
distribution. Find the mode of each of the following distributions:
(a) \(p(x)=\left(\frac{1}{2}\right)^{x}, x=1,2,3, \ldots\), zero elsewhere.
(b) \(f(x)=12 x^{2}(1-x), 0
In a certain factory, machines I, II, and III are all producing springs of the same length. Machines I, II, and III produce \(1 \%, 4 \%\) and \(2 \%\) defective springs, respectively. Of the total production of springs in the factory, Machine I produces \(30 \%\), Machine II produces \(25 \%\), and Machine III produces \(45 \%\). (a) If one spring is selected at random from the total springs produced in a given day, determine the probability that it is defective. (b) Given that the selected spring is defective, find the conditional probability that it was produced by Machine II.
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