Create a Chi-Square distribution
Chi-square distributions show up often in frequentist settings as the sampling distribution of test statistics, especially in maximum likelihood estimation settings.
ChiSquare(df)
df |
Degrees of freedom. Must be positive. |
We recommend reading this documentation on https://alexpghayes.github.io/distributions3, where the math will render with additional detail and much greater clarity.
In the following, let X be a χ^2 random variable with
df = k.
Support: R^+, the set of positive real numbers
Mean: k
Variance: 2k
Probability density function (p.d.f):
f(x) = 1 / (2 π σ^2) exp(-(x - μ)^2 / (2 σ^2))
Cumulative distribution function (c.d.f):
The cumulative distribution function has the form
F(t) = integral_{-∞}^t 1 / (2 π σ^2) exp(-(x - μ)^2 / (2 σ^2)) dx
but this integral does not have a closed form solution and must be approximated numerically. The c.d.f. of a standard normal is sometimes called the "error function". The notation Φ(t) also stands for the c.d.f. of a standard normal evaluated at t. Z-tables list the value of Φ(t) for various t.
Moment generating function (m.g.f):
E(e^(tX)) = e^(μ t + σ^2 t^2 / 2)
A ChiSquare object.
A squared standard Normal() distribution is equivalent to a
χ^2_1 distribution with one degree of freedom. The
χ^2 distribution is a special case of the Gamma()
distribution with shape (TODO: check this) parameter equal
to a half. Sums of χ^2 distributions
are also distributed as χ^2 distributions, where the
degrees of freedom of the contributing distributions get summed.
The ratio of two χ^2 distributions is a FisherF()
distribution. The ratio of a Normal() and the square root
of a scaled ChiSquare() is a StudentsT() distribution.
set.seed(27) X <- ChiSquare(5) X random(X, 10) pdf(X, 2) log_pdf(X, 2) cdf(X, 4) quantile(X, 0.7) cdf(X, quantile(X, 0.7)) quantile(X, cdf(X, 7))
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