Continuous Probability Function

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A Continuous Probability Function is a probability distribution function that can accept a continuous random variable.

References

2015

1. Probability distribution function PlanetMath
2. Probability Function at Mathworld
3. Ord, J.K. (1972) Families of Frequency Distributions, Griffin. ISBN 0-85264-137-0 (for example, Table 5.1 and Example 5.4)

2000

• (Valpola, 2000) ⇒ Harri Valpola. (2000). “Bayesian Ensemble Learning for Nonlinear Factor Analysis." PhD Dissertation, Helsinki University of Technology.
• QUOTE: probability density: Any single value of a continuous valued variable usually has zero probability and only a finite range of values has a nonzero probability. Probability of a continuous variable can be characterised by probability density which is defined to be the probability of a range divided by the size of the range.
• QUOTE: volume: In analogy to physical mass, density and volume, the size of range of continuous valued variables can be called volume. See probability density.

1986

• (Larsen & Marx, 1986) ⇒ Richard J. Larsen, and Morris L. Marx. (1986). “An Introduction to Mathematical Statistics and Its Applications, 2nd edition." Prentice Hall
• 'Definition 3.2.1. A real-valued function whose domain is the sample space S is called a random variable. We denote random variables by uppercase letters, often $\displaystyle{ X }$, Y, or Z.

If the range of the mapping contains either a finite or countably infinite number of values, the random variable is said to be discrete ; if the range includes an interval of real numbers, bounded or unbounded, the random variable is said to be continuous.

Associated with each discrete random variable $\displaystyle{ Y }$ is a probability density function (or pdf). “fY(y). By definition, fY(y) is the sum of all the probabilities associated with outcomes in $\displaystyle{ S }$ that get mapped into $\displaystyle{ y }$ by the random variable Y. That is.

• fY(y) = P({s(∈)S |Y(s) = y})
• Conceptually, fY(y) describes the probability structure induced on the real line by the random variable Y.

For notational simplicity, we will delete all references to $\displaystyle{ s }$ and $\displaystyle{ S }$ and write: fY(y) = P(Y(s)=y). In other words, fY(y) is the "probability that the random variable Y takes on the value y."

Associated with each continuous random variable $\displaystyle{ Y }$ is also a probability density function, fY(y), but fY(y) in this case is not the probability that the random variable $\displaystyle{ Y }$ takes on the value y. Rather, fY(y) is a continuous curve having the property that for all $\displaystyle{ a }$ and $\displaystyle{ b }$,

• P(a$\displaystyle{ Y }$b) = P({s(∈)S| $\displaystyle{ a }$Y(s) ≤ b}) = Integral(a,b). “fY(y) dy]

1972

• (Ord, 1972) ⇒ J. K. Ord. (1972). “Families of Frequency Distributions.” Griffin. ISBN:0-85264-137-0