| Chisquare {base} | R Documentation |
The (non-central) Chi-Squared Distribution
Description
Density, distribution function, quantile function and random
generation for the chi-squared (\chi^2) distribution with
df degrees of freedom and optional non-centrality parameter
ncp.
Usage
dchisq(x, df, ncp=0, log = FALSE)
pchisq(q, df, ncp=0, lower.tail = TRUE, log.p = FALSE)
qchisq(p, df, ncp=0, lower.tail = TRUE, log.p = FALSE)
rchisq(n, df, ncp=0)
Arguments
x, q |
vector of quantiles. |
p |
vector of probabilities. |
n |
number of observations. If |
df |
degrees of freedom (positive, but can be non-integer). |
ncp |
non-centrality parameter. For |
log, log.p |
logical; if TRUE, probabilities p are given as log(p). |
lower.tail |
logical; if TRUE (default), probabilities are
|
Details
The chi-squared distribution with df= n degrees of
freedom has density
f_n(x) = \frac{1}{{2}^{n/2} \Gamma (n/2)} {x}^{n/2-1} {e}^{-x/2}
for x > 0. The mean and variance are n and 2n.
The non-central chi-squared distribution with df= n
degrees of freedom and non-centrality parameter ncp
= \lambda has density
f(x) = e^{-\lambda / 2}
\sum_{r=0}^\infty \frac{(\lambda/2)^r}{r!}\, f_{n + 2r}(x)
for x \ge 0. It is the distribution of the sum of squares of
n normals each with variance one, \lambda being the sum of
squares of the normal means.
Value
dchisq gives the density, pchisq gives the distribution
function, qchisq gives the quantile function, and rchisq
generates random deviates.
See Also
dgamma for the Gamma distribution which generalizes the
chi-squared one.
Examples
dchisq(1, df=1:3)
pchisq(1, df= 3)
pchisq(1, df= 3, ncp = 0:4)# includes the above
x <- 1:10
## Chi-squared(df = 2) is a special exponential distribution
all.equal(dchisq(x, df=2), dexp(x, 1/2))
all.equal(pchisq(x, df=2), pexp(x, 1/2))