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This distribution for a = 0, b = 1 and c = 0.5—the mode (i.e., the peak) is exactly in the middle of the interval—corresponds to the distribution of the mean of two standard uniform variables, that is, the distribution of X = (X 1 + X 2) / 2, where X 1, X 2 are two independent random variables with standard uniform distribution in [0, 1]. [1]
The probability of default is an estimate of the likelihood that the default event will occur. It applies to a particular assessment horizon, usually one year. Credit scores, such as FICO for consumers or bond ratings from S&P, Fitch or Moodys for corporations or governments, typically imply a certain probability of default.
In case (1) above, the conditional probability that the longitude λ lies in a set E given that φ = 0 can be written P(λ ∈ E | φ = 0). Elementary probability theory suggests this can be computed as P(λ ∈ E and φ = 0)/P(φ = 0), but that expression is not well-defined since P(φ = 0) = 0.
The Bertrand paradox is a problem within the classical interpretation of probability theory. Joseph Bertrand introduced it in his work Calcul des probabilités (1889) [1] as an example to show that the principle of indifference may not produce definite, well-defined results for probabilities if it is applied uncritically when the domain of possibilities is infinite.
The probability distribution of the sum of two or more independent random variables is the convolution of their individual distributions. The term is motivated by the fact that the probability mass function or probability density function of a sum of independent random variables is the convolution of their corresponding probability mass functions or probability density functions respectively.
In probability theory, Kolmogorov's zero–one law, named in honor of Andrey Nikolaevich Kolmogorov, specifies that a certain type of event, namely a tail event of independent σ-algebras, will either almost surely happen or almost surely not happen; that is, the probability of such an event occurring is zero or one.
The Machina Triangle is a way of representing a three dimensional probability vector in a two dimensional space. The probability of a given outcome is denoted by a euclidean distance from the point that represents a lottery (probability). [1]
The probability is sometimes written to distinguish it from other functions and measure P to avoid having to define "P is a probability" and () is short for ({: ()}), where is the event space, is a random variable that is a function of (i.e., it depends upon ), and is some outcome of interest within the domain specified by (say, a particular ...