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In probability theory, a logit-normal distribution is a probability distribution of a random variable whose logit has a normal distribution.If Y is a random variable with a normal distribution, and t is the standard logistic function, then X = t(Y) has a logit-normal distribution; likewise, if X is logit-normally distributed, then Y = logit(X)= log (X/(1-X)) is normally distributed.
Example: To find 0.69, one would look down the rows to find 0.6 and then across the columns to 0.09 which would yield a probability of 0.25490 for a cumulative from mean table or 0.75490 from a cumulative table. To find a negative value such as -0.83, one could use a cumulative table for negative z-values [3] which yield a probability of 0.20327.
If X is a gamma(α, β) random variable and the shape parameter α is large relative to the scale parameter β, then X approximately has a normal random variable with the same mean and variance. If X is a Student's t random variable with a large number of degrees of freedom ν then X approximately has a standard normal distribution.
The parameter is in this case imaginary, but the function is nevertheless real, positive, and normalizable. The scale parameter of the untruncated normal distribution must be positive because the distribution would not be normalizable otherwise. The doubly truncated normal distribution, on the other hand, can in principle have a negative scale ...
The simplest case of a normal distribution is known as the standard normal distribution or unit normal distribution. This is a special case when μ = 0 {\textstyle \mu =0} and σ 2 = 1 {\textstyle \sigma ^{2}=1} , and it is described by this probability density function (or density): φ ( z ) = e − z 2 2 2 π . {\displaystyle \varphi (z ...
Extreme values like maximum one-day rainfall and river discharge per month or per year often follow a log-normal distribution. [12] The log-normal distribution, however, needs a numeric approximation. As the log-logistic distribution, which can be solved analytically, is similar to the log-normal distribution, it can be used instead.
The half-normal distribution is a special case of the generalized gamma distribution with d = 1, p = 2, a = . If Y has a half-normal distribution, Y-2 has a Lévy distribution; The Rayleigh distribution is a moment-tilted and scaled generalization of the half-normal distribution.
In probability and statistics, the logarithmic distribution (also known as the logarithmic series distribution or the log-series distribution) is a discrete probability distribution derived from the Maclaurin series expansion