I'm trying to learn how the names density and distribution, and related terms, are used in statistics and probability theory. Here are four concepts which I've labelled by pairs:(adsbygoogle = window.adsbygoogle || []).push({});

[tex](0, 0) = f:\mathbb{R} \to [0,1] \; : \; f(x)=P(X=x)=P(\{s \in S : X(s)=x\})[/tex]

where s is an element of a sample space, S.

[tex](0, 1) = F(x)=P(X \leq x)=\sum_{t=-\infty}^{x}f(t)[/tex]

[tex](1, 0) = g(x) : [/tex]

[tex](i) \, g(x) \geq 0;[/tex]

[tex](ii) \, \int_{-\infty}^{\infty}g(x) \, dx = 1;[/tex]

[tex](iii) \, \int_{a}^{b} g(x) \, dx = P(a < X < b)[/tex]

where a and b are any two values of X such that a < b.

[tex](1, 1) = G(x)=P(X\leq x) = \int_{-\infty}^{x}g(t) \, dt[/tex]

Hoel, inIntroduction to Mathematical Statistics, calls concepts (n, 0) probability densities. He calls concepts (n, 1) probability distributions. Wolfram Mathworld gives the same definitions, but adds that some people use the term "cumulativedistribution function", CDF, in place of probability distribution. Wikipedia calls (0, 0) a probability mass function (only discrete), (1, 0) a probability density function (only continuous), and (n, 1) cumulative distribution functions, discrete and continuous.

However, Mathworld seems to use the term distribution in a different sense with reference to, for example, the binomial and normal "distributions". The formula they define the binomial distribution by here [ http://mathworld.wolfram.com/BinomialDistribution.html ] is called a "probability density function" by Wolfram Alpha and a "probability mass function" by Wikipedia. Hoel also seems to neglect his earlier definitions and use the terms distribution and density interchangeably with respect to the binomial and normal distribution (or density).

I get the impression that for everyone in the context of the binomial and normal distributions, and for Wikipedia in general, "(probability) distribution" may refer to something broader than (Inclusive of? Related to?) Hoel's densities and distributions, so that the bell curve is only the graph of the probability density functionassociated withthe normal distribution, and neither this density nor the corresponding cumulative distribution function are to be identified as the normal distribution itself. Is that right? Could someone explain more exactly what distribution is here?

I'm afraid I don't understand enough of the supporting terminology to understand Wikipedia's definition of probability distribution [ http://en.wikipedia.org/wiki/Probability_distribution ], but I notice that it avoids identifying it with either probability mass function / probability density, or cumulative distribution function. Instead it just says that probability distributions are "characterized by" these. What exactly is the relationship?

This article uses the notation Pr[X = x] and Pr[X < or = x]. So their probability distribution seems to be a function of two variables, Pr(x, r), where x is a real number, and r is one of two relations, either "=" or "< or =", so that, depending on the relation, it could manifest as Hoel's/Mathworld's density, or Hoel's/Mathworld's (cumulative) distribution. Is that anywhere near the mark?

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# Density and distribution

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