| StudentsT {distributions3} | R Documentation |
Create a Student's T distribution
Description
The Student's T distribution is closely related to the Normal()
distribution, but has heavier tails. As \nu increases to \infty,
the Student's T converges to a Normal. The T distribution appears
repeatedly throughout classic frequentist hypothesis testing when
comparing group means.
Usage
StudentsT(df)
Arguments
df |
Degrees of freedom. Can be any positive number. Often
called |
Details
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 Students T random variable with
df = \nu.
Support: R, the set of all real numbers
Mean: Undefined unless \nu \ge 2, in which case the mean is
zero.
Variance:
\frac{\nu}{\nu - 2}
Undefined if \nu < 1, infinite when 1 < \nu \le 2.
Probability density function (p.d.f):
f(x) = \frac{\Gamma(\frac{\nu + 1}{2})}{\sqrt{\nu \pi} \Gamma(\frac{\nu}{2})} (1 + \frac{x^2}{\nu} )^{- \frac{\nu + 1}{2}}
Cumulative distribution function (c.d.f):
Nasty, omitted.
Moment generating function (m.g.f):
Undefined.
Value
A StudentsT object.
See Also
Other continuous distributions:
Beta(),
Cauchy(),
ChiSquare(),
Erlang(),
Exponential(),
FisherF(),
Frechet(),
GEV(),
GP(),
Gamma(),
Gumbel(),
LogNormal(),
Logistic(),
Normal(),
RevWeibull(),
Tukey(),
Uniform(),
Weibull()
Examples
set.seed(27)
X <- StudentsT(3)
X
random(X, 10)
pdf(X, 2)
log_pdf(X, 2)
cdf(X, 4)
quantile(X, 0.7)
### example: calculating p-values for two-sided T-test
# here the null hypothesis is H_0: mu = 3
# data to test
x <- c(3, 7, 11, 0, 7, 0, 4, 5, 6, 2)
nx <- length(x)
# calculate the T-statistic
t_stat <- (mean(x) - 3) / (sd(x) / sqrt(nx))
t_stat
# null distribution of statistic depends on sample size!
T <- StudentsT(df = nx - 1)
# calculate the two-sided p-value
1 - cdf(T, abs(t_stat)) + cdf(T, -abs(t_stat))
# exactly equivalent to the above
2 * cdf(T, -abs(t_stat))
# p-value for one-sided test
# H_0: mu <= 3 vs H_A: mu > 3
1 - cdf(T, t_stat)
# p-value for one-sided test
# H_0: mu >= 3 vs H_A: mu < 3
cdf(T, t_stat)
### example: calculating a 88 percent T CI for a mean
# lower-bound
mean(x) - quantile(T, 1 - 0.12 / 2) * sd(x) / sqrt(nx)
# upper-bound
mean(x) + quantile(T, 1 - 0.12 / 2) * sd(x) / sqrt(nx)
# equivalent to
mean(x) + c(-1, 1) * quantile(T, 1 - 0.12 / 2) * sd(x) / sqrt(nx)
# also equivalent to
mean(x) + quantile(T, 0.12 / 2) * sd(x) / sqrt(nx)
mean(x) + quantile(T, 1 - 0.12 / 2) * sd(x) / sqrt(nx)