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In the absolutely continuous case, probabilities are described by a probability density function, and the probability distribution is by definition the integral of the probability density function. [7] [4] [8] The normal distribution is a commonly encountered absolutely continuous probability distribution.
The probability density function is nonnegative everywhere, and the area under the entire curve is equal to 1. The terms probability distribution function and probability function have also sometimes been used to denote the probability density function. However, this use is not standard among probabilists and statisticians.
If () is a general scalar-valued function of a normal vector, its probability density function, cumulative distribution function, and inverse cumulative distribution function can be computed with the numerical method of ray-tracing (Matlab code). [17]
The probability density, cumulative distribution, and inverse cumulative distribution of any function of one or more independent or correlated normal variables can be computed with the numerical method of ray-tracing [41] (Matlab code). In the following sections we look at some special cases.
a function of t, determines the behavior and properties of the probability distribution of X. It is equivalent to a probability density function or cumulative distribution function, since knowing one of these functions allows computation of the others, but they provide different insights into the features of the random variable. In particular ...
In probability theory, a log-normal (or lognormal) distribution is a continuous probability distribution of a random variable whose logarithm is normally distributed. Thus, if the random variable X is log-normally distributed, then Y = ln( X ) has a normal distribution.
The gamma distribution is the maximum entropy probability distribution (both with respect to a uniform base measure and a / base measure) for a random variable X for which E[X] = αθ = α/λ is fixed and greater than zero, and E[ln X] = ψ(α) + ln θ = ψ(α) − ln λ is fixed (ψ is the digamma function). [5]
The function A(t | ν) is the integral of Student's probability density function, f(t) between -t and t, for t ≥ 0 . It thus gives the probability that a value of t less than that calculated from observed data would occur by chance.