(25th-August-2020)
The definition of probability is sometimes specified in terms of probability density functions when the domain is continuous (e.g., a subset of the real numbers). A probability density function provides a way to give a measure over sets of possible worlds. The measure is defined in terms of an integral of a probability density function. The formal definition of an integral is the limit of the discretizations as the discretizations become finer.
The only non-discrete probability distribution we will use in this book is where the domain is the real line. In this case, there is a possible world for each real number. A probability density function, which we write as p, is a function from reals into non-negative reals that integrates to 1. The probability that a real-valued random variable X has value between a and b is given by
P(a ≤ X ≤ b)=∫ab p(X ) dX .
A parametric distribution is one where the density function can be described by a formula. Although not all distributions can be described by formulas, all of the ones that we can represent are. Sometimes statisticians use the term parametric to mean the distribution can be described using a fixed, finite number of parameters. A non-parametric distribution is one where the number of parameters is not fixed. (Oddly, non-parametric typically means "many parameters").
Regarding axioms provability,
• The preceding section gave a semantic definition of probability. We can also give an axiomatic definition of probability that specifies axioms of what properties we may want in a calculus of belief. Suppose P is a function from propositions into real numbers that satisfies the following three axioms of probability:
• Axiom 1
• 0 ≤ P(α) for any proposition α. That is, the belief in any proposition cannot be negative.
• Axiom 2
• P(τ) = 1 if τ is a tautology. That is, if τ is true in all possible worlds, its probability is 1.
• Axiom 3
• P(α∨β)=P(α)+P(β) if α and β are contradictory propositions; that is, if ¬(α∧β) is a tautology. In other words, if two propositions cannot both be true (they are mutually exclusive), the probability of their disjunction is the sum of their probabilities.
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