Uniform Distribution

The Uniform distribution is an absolute continuous probability distribution where all intervals of the same length within the distribution's support are equally probable. It is defined by two parameters: the lower bound \(a\) and the upper bound \(b\), with \(a < b\).

Usage

Unif(min = 0, max = 1)

# S4 method for class 'Unif,numeric'
d(distr, x, log = FALSE)

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

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

# S4 method for class 'Unif,numeric'
r(distr, n)

# S4 method for class 'Unif'
mean(x)

# S4 method for class 'Unif'
median(x)

# S4 method for class 'Unif'
mode(x)

# S4 method for class 'Unif'
var(x)

# S4 method for class 'Unif'
sd(x)

# S4 method for class 'Unif'
skew(x)

# S4 method for class 'Unif'
kurt(x)

# S4 method for class 'Unif'
entro(x)

llunif(x, min, max)

# S4 method for class 'Unif,numeric'
ll(distr, x)

eunif(x, type = "mle", ...)

# S4 method for class 'Unif,numeric'
mle(distr, x, na.rm = FALSE)

# S4 method for class 'Unif,numeric'
me(distr, x, na.rm = FALSE)

Arguments

min, max

numeric. The distribution parameters.

distr

an object of class Unif.

x

For the density function, x is a numeric vector of quantiles. For the moments functions, x is an object of class Unif. 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.

type

character, case ignored. The estimator type (mle or me).

...

extra arguments.

na.rm

logical. Should the NA values be removed?

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.

Details

The probability density function (PDF) of the Uniform distribution is: $$ f(x; a, b) = \frac{1}{b - a}, \quad a \le x \le b .$$

See also

Functions from the stats package: dunif(), punif(), qunif(), runif()

Examples

# -----------------------------------------------------
# Uniform Distribution Example
# -----------------------------------------------------

# Create the distribution
a <- 3 ; b <- 5
D <- Unif(a, b)

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

d(D, c(0.3, 0.8, 0.5)) # density function
#> [1] 0 0 0
p(D, c(0.3, 0.8, 0.5)) # distribution function
#> [1] 0 0 0
qn(D, c(0.4, 0.8)) # inverse distribution function
#> [1] 3.8 4.6
x <- r(D, 100) # random generator function

# alternative way to use the function
df <- d(D) ; df(x) # df is a function itself
#>   [1] 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5
#>  [19] 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5
#>  [37] 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5
#>  [55] 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5
#>  [73] 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5
#>  [91] 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5

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

mean(D) # Expectation
#> [1] 4
var(D) # Variance
#> [1] 0.3333333
sd(D) # Standard Deviation
#> [1] 0.5773503
skew(D) # Skewness
#> [1] 0
kurt(D) # Excess Kurtosis
#> [1] -1.2
entro(D) # Entropy
#> [1] 0.6931472

# List of all available moments
mom <- moments(D)
#> Warning: The mode is any element in the support (or its interior) of
#>             a Uniform distribution. The mean is returned by default.
mom$mean # expectation
#> [1] 4

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

ll(D, x)
#> [1] -69.31472
llunif(x, a, b)
#> [1] -69.31472

eunif(x, type = "mle")
#> $min
#> [1] 3.025647
#> 
#> $max
#> [1] 4.996425
#> 
eunif(x, type = "me")
#> $min
#> [1] 2.9922
#> 
#> $max
#> [1] 5.056657
#> 

mle(D, x)
#> $min
#> [1] 3.025647
#> 
#> $max
#> [1] 4.996425
#> 
me(D, x)
#> $min
#> [1] 2.9922
#> 
#> $max
#> [1] 5.056657
#> 
e(D, x, type = "mle")
#> $min
#> [1] 3.025647
#> 
#> $max
#> [1] 4.996425
#> 

mle("unif", x) # the distr argument can be a character
#> $min
#> [1] 3.025647
#> 
#> $max
#> [1] 4.996425
#>