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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 ...
In probability theory and statistics, the F-distribution or F-ratio, also known as Snedecor's F distribution or the Fisher–Snedecor distribution (after Ronald Fisher and George W. Snedecor), is a continuous probability distribution that arises frequently as the null distribution of a test statistic, most notably in the analysis of variance (ANOVA) and other F-tests.
If, on the other hand, we know the characteristic function φ and want to find the corresponding distribution function, then one of the following inversion theorems can be used. Theorem. If the characteristic function φ X of a random variable X is integrable, then F X is absolutely continuous, and therefore X has a probability density function.
The formula for the one-way ANOVA F-test statistic is =, or =. The "explained variance", or "between-group variability" is = (¯ ¯) / where ¯ denotes the sample mean in the i-th group, is the number of observations in the i-th group, ¯ denotes the overall mean of the data, and denotes the number of groups.
This is also called a "change of variable" and is in practice used to generate a random variable of arbitrary shape f g(X) = f Y using a known (for instance, uniform) random number generator. It is tempting to think that in order to find the expected value E(g(X)), one must first find the probability density f g(X) of the new random variable Y ...
A probability distribution can be described in various forms, such as by a probability mass function or a cumulative distribution function. One of the most general descriptions, which applies for absolutely continuous and discrete variables, is by means of a probability function : whose input space is a σ-algebra, and gives a real number ...
A vector X ∈ R k is multivariate-normally distributed if any linear combination of its components Σ k j=1 a j X j has a (univariate) normal distribution. The variance of X is a k×k symmetric positive-definite matrix V. The multivariate normal distribution is a special case of the elliptical distributions.
For k > 1, the density function tends to zero as x approaches zero from above, increases until its mode and decreases after it. The density function has infinite negative slope at x = 0 if 0 < k < 1, infinite positive slope at x = 0 if 1 < k < 2 and null slope at x = 0 if k > 2. For k = 1 the density has a finite negative slope at x = 0.