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In statistics, the variance function is a smooth function that depicts the variance of a random quantity as a function of its mean.The variance function is a measure of heteroscedasticity and plays a large role in many settings of statistical modelling.
The probability function, mean and variance are given in the adjacent table. An alternative expression of the distribution has both the number of balls taken of each color and the number of balls not taken as random variables, whereby the expression for the probability becomes symmetric.
In probability theory, it is possible to approximate the moments of a function f of a random variable X using Taylor expansions, provided that f is sufficiently differentiable and that the moments of X are finite.
This algorithm can easily be adapted to compute the variance of a finite population: simply divide by n instead of n − 1 on the last line.. Because SumSq and (Sum×Sum)/n can be very similar numbers, cancellation can lead to the precision of the result to be much less than the inherent precision of the floating-point arithmetic used to perform the computation.
The variance then is expressed easily in terms of the mean: = +. Both the mean (μ) and variance (σ 2) of X in the original normal distribution can be interpreted as the location and scale parameters of Y in the folded distribution.
It is also the continuous distribution with the maximum entropy for a specified mean and variance. [18] [19] Geary has shown, assuming that the mean and variance are finite, that the normal distribution is the only distribution where the mean and variance calculated from a set of independent draws are independent of each other. [20] [21]
An exGaussian random variable Z may be expressed as Z = X + Y, where X and Y are independent, X is Gaussian with mean μ and variance σ 2, and Y is exponential of rate λ. It has a characteristic positive skew from the exponential component.
In statistics, the Q-function is the tail distribution function of the standard normal distribution. [ 1 ] [ 2 ] In other words, Q ( x ) {\displaystyle Q(x)} is the probability that a normal (Gaussian) random variable will obtain a value larger than x {\displaystyle x} standard deviations.