Stud: Student Distribution

In joker: Probability Distributions and Parameter Estimation

Student Distribution

Description

The Student's t-distribution is a continuous probability distribution used primarily in hypothesis testing and in constructing confidence intervals for small sample sizes. It is defined by one parameter: the degrees of freedom \nu > 0.

Usage

Stud(df = 1)

## S4 method for signature 'Stud,numeric'
d(distr, x, log = FALSE)

## S4 method for signature 'Stud,numeric'
p(distr, q, lower.tail = TRUE, log.p = FALSE)

## S4 method for signature 'Stud,numeric'
qn(distr, p, lower.tail = TRUE, log.p = FALSE)

## S4 method for signature 'Stud,numeric'
r(distr, n)

## S4 method for signature 'Stud'
mean(x)

## S4 method for signature 'Stud'
median(x)

## S4 method for signature 'Stud'
mode(x)

## S4 method for signature 'Stud'
var(x)

## S4 method for signature 'Stud'
sd(x)

## S4 method for signature 'Stud'
skew(x)

## S4 method for signature 'Stud'
kurt(x)

## S4 method for signature 'Stud'
entro(x)

llt(x, df)

## S4 method for signature 'Stud,numeric'
ll(distr, x)

Arguments

df	numeric. The distribution degrees of freedom parameter.
distr	an object of class Stud.
x	For the density function, x is a numeric vector of quantiles. For the moments functions, x is an object of class Stud. For the log-likelihood and the estimation functions, x is the sample of observations.
log, log.p	logical. Should the logarithm of the probability be returned?
q	numeric. Vector of quantiles.
lower.tail	logical. If TRUE (default), probabilities are P(X \leq x), otherwise P(X > x).
p	numeric. Vector of probabilities.
n	number of observations. If length(n) > 1, the length is taken to be the number required.

Details

The probability density function (PDF) of the Student's t-distribution is:

f(x; \nu) = \frac{\Gamma\left(\frac{\nu + 1}{2}\right)}{\sqrt{\nu\pi}\ \Gamma\left(\frac{\nu}{2}\right)}\left(1 + \frac{x^2}{\nu}\right)^{-\frac{\nu + 1}{2}} .

Value

Each type of function returns a different type of object:

• Distribution Functions: When supplied with one argument (distr), the d(), p(), q(), r(), ll() functions return the density, cumulative probability, quantile, random sample generator, and log-likelihood functions, respectively. When supplied with both arguments (distr and x), they evaluate the aforementioned functions directly.

• Moments: Returns a numeric, either vector or matrix depending on the moment and the distribution. The moments() function returns a list with all the available methods.

• Estimation: Returns a list, the estimators of the unknown parameters. Note that in distribution families like the binomial, multinomial, and negative binomial, the size is not returned, since it is considered known.

• Variance: Returns a named matrix. The asymptotic covariance matrix of the estimator.

See Also

Functions from the stats package: dt(), pt(), qt(), rt()

Examples

# -----------------------------------------------------
# Student Distribution Example
# -----------------------------------------------------

# Create the distribution
df <- 12
D <- Stud(df)

# ------------------
# dpqr Functions
# ------------------

d(D, c(-3, 0, 3)) # density function
p(D, c(-3, 0, 3)) # distribution function
qn(D, c(0.4, 0.8)) # inverse distribution function
x <- r(D, 100) # random generator function

# alternative way to use the function
d1 <- d(D) ; d1(x) # d1 is a function itself

# ------------------
# Moments
# ------------------

mean(D) # Expectation
median(D) # Median
mode(D) # Mode
var(D) # Variance
sd(D) # Standard Deviation
skew(D) # Skewness
kurt(D) # Excess Kurtosis
entro(D) # Entropy

# List of all available moments
mom <- moments(D)
mom$mean # expectation

# ------------------
# Point Estimation
# ------------------

ll(D, x)
llt(x, df)

joker documentation built on June 8, 2025, 12:12 p.m.