Computing the distribution of the product of two continuous random variables

Glen, Andrew G.; Leemis, Lawrence M.; Drew, John H.

doi:10.1016/s0167-9473(02)00234-7

Cited by 124 publications

(75 citation statements)

References 3 publications

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“…Expression (30) shows that regressing average earnings on population, while controlling for talent, yields an estimate of agglomeration economies, ε. Now, using the equilibrium relationship linking city population sizes to the distribution of talent (19), L γ−ε c = ξ((1 + γ)/(1 + ε))t 1+a c , to control for the shift we get:…”

Section: Quantitative Implicationsmentioning

confidence: 99%

Productive Cities: Sorting, Selection, and Agglomeration

Behrens

Duranton

Robert‐Nicoud

2014

Journal of Political Economy

330

226

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Large cities produce more output per capita than small cities. This higher productivity may occur because more talented individuals sort into large cities, because large cities select more productive entrepreneurs and firms, or because of agglomeration economies. We develop a model of systems of cities that combines all three elements and suggests interesting complementarities between them. The model can replicate stylized facts about sorting, agglomeration, and selection in cities. It also generates Zipf 's law for cities under empirically plausible parameter values. Finally, it provides a useful framework within which to reinterpret extant empirical evidence. Abstract Large cities produce more output per capita than small cities. This may occur because more talented individuals sort into large cities, because large cities select more productive entrepreneurs and firms, or because of agglomeration economies. We develop a model of systems of cities that combines all three elements and suggests interesting complementarities between them. The model can replicate stylised facts about sorting, agglomeration, and selection in cities. It also generates Zipf's law for cities under empirically plausible parameter values. Finally, it provides a useful framework within which to reinterpret extant empirical evidence.

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Section: Quantitative Implicationsmentioning

confidence: 99%

Productive Cities: Sorting, Selection, and Agglomeration

Behrens

Duranton

Robert‐Nicoud

2014

Journal of Political Economy

330

226

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“…The problem of finding the PDF of the product of two random variables was first solved by Rohatgi (1976) but its implementation relied on knowledge of the joint PDF of the two variables. In this paper, we apply the approach followed by Glen et al (2004), which uses the individual PDFs of the two random variables.…”

Section: The Problem and Our Conceptmentioning

confidence: 99%

The Effects of Viewing Angle on the Mass Distribution of Exoplanets

López

Jenkins

2012

ApJ

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We present a mathematical method to statistically decouple the effects of unknown inclination angles on the mass distribution of exoplanets that have been discovered using radial-velocity (RV) techniques. The method is based on the distribution of the product of two random variables. Thus, if one assumes a true mass distribution, the method makes it possible to recover the observed distribution. We compare our prediction with available RV data. Assuming that the true mass function is described by a power law, the minimum mass function that we recover proves a good fit to the observed distribution at both mass ends. In particular, it provides an alternative explanation for the observed low-mass decline, usually explained as sample incompleteness. In addition, the peak observed near the low-mass end arises naturally in the predicted distribution as a consequence of imposing a low-mass cutoff in the true distribution. If the low-mass bins below 0.02 M J are complete, then the mass distribution in this regime is heavily affected by the small fraction of lowly inclined interlopers that are actually more massive companions. Finally, we also present evidence that the exoplanet mass distribution changes form toward low mass, implying that a single power law may not adequately describe the sample population.

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“…v i is the product of two standard normal random variables, which follows an interesting distribution discussed in [12]. The plot of this distribution is shown in figure 2.…”

Section: Kernel Behavior In High Dimensional Input Spacementioning

confidence: 95%

Taming the Curse of Dimensionality in Kernels and Novelty Detection

Evangelista

Embrechts

Szymanski

Advances in Soft Computing

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Abstract. The curse of dimensionality is a well known but not entirely well-understood phenomena. Too much data, in terms of the number of input variables, is not always a good thing. This is especially true when the problem involves unsupervised learning or supervised learning with unbalanced data (many negative observations but minimal positive observations). This paper addresses two issues involving high dimensional data: The first issue explores the behavior of kernels in high dimensional data. It is shown that variance, especially when contributed by meaningless noisy variables, confounds learning methods. The second part of this paper illustrates methods to overcome dimensionality problems with unsupervised learning utilizing subspace models. The modeling approach involves novelty detection with the one-class SVM. IntroductionHigh dimensional data often create problems. This problem is exacerbated if the training data is only one class, unknown classes, or significantly unbalanced classes. Consider a binary classification problem that involves computer intrusion detection. Our intention is to classify network traffic, and we are interested in classifying the traffic as either attacks (intruders) or non attacks. Capturing network traffic is simple -hookup to a LAN cable, run tcpdump, and you can fill a hard drive within minutes. These captured network connections can be described with attributes; it is not uncommon for a network connection to be described with over 100 attributes [14]. However, the class of each connection will be unknown, or perhaps with reasonable confidence we can assume that all of the connections do not involve any attacks. The above scenario can be generalized to other security problems as well. Given a matrix of data, X, containing N observations and m attributes, we are interested in classifying this data as either potential attackers (positive class) or non attackers (negative class). If m is large, and our labels, y ∈ R N ×1 , are unbalanced (usually plenty of known non attackers and few instances of attacks), one class (all non attackers), or unknown, increased dimensionality rapidly becomes a problem and feature selection is not feasible due to the minimal examples (if any) of the attacker class.

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Computing the distribution of the product of two continuous random variables

Cited by 124 publications

References 3 publications

Productive Cities: Sorting, Selection, and Agglomeration

Productive Cities: Sorting, Selection, and Agglomeration

The Effects of Viewing Angle on the Mass Distribution of Exoplanets

Taming the Curse of Dimensionality in Kernels and Novelty Detection

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