Bern Distribution

The Bernoulli distribution is a discrete probability distribution which takes the value 1 with probability $p$ and the value 0 with probability $1 - p$, where $0 \leq p \leq 1$.

Usage

Bern(prob = 0.5)

dbern(x, prob, log = FALSE)

pbern(q, prob, lower.tail = TRUE, log.p = FALSE)

qbern(p, prob, lower.tail = TRUE, log.p = FALSE)

rbern(n, prob)

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

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

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

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

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

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

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

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

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

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

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

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

# S4 method for class 'Bern'
finf(x)

llbern(x, prob)

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

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

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

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

vbern(prob, type = "mle")

# S4 method for class 'Bern'
avar_mle(distr)

# S4 method for class 'Bern'
avar_me(distr)

Arguments

prob: numeric. Probability of success.
x: For the density function, x is a numeric vector of quantiles. For the moments functions, x is an object of class Bern. 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.
distr: an object of class Bern.
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 mass function (PMF) of the Bernoulli distribution is given by: $$ f(x; p) = p^x (1 - p)^{1 - x}, \quad p \in (0, 1), \quad x \in \{0, 1\}.$$

Examples

# -----------------------------------------------------
# Bernoulli Distribution Example
# -----------------------------------------------------

# Create the distribution
p <- 0.7
D <- Bern(p)

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

d(D, c(0, 1)) # density function
#> [1] 0.3 0.7
p(D, c(0, 1)) # distribution function
#> [1] 0.3 1.0
qn(D, c(0.4, 0.8)) # inverse distribution function
#> [1] 1 1
x <- r(D, 100) # random generator function

# alternative way to use the function
df <- d(D) ; df(x) # df is a function itself
#>   [1] 0.7 0.3 0.7 0.7 0.7 0.7 0.7 0.7 0.3 0.3 0.3 0.7 0.7 0.7 0.7 0.7 0.7 0.7
#>  [19] 0.7 0.3 0.7 0.7 0.3 0.7 0.3 0.3 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7
#>  [37] 0.3 0.3 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.3 0.3 0.3 0.3 0.7 0.7 0.7 0.7 0.7
#>  [55] 0.7 0.3 0.7 0.3 0.7 0.7 0.3 0.7 0.7 0.7 0.7 0.7 0.3 0.7 0.7 0.7 0.7 0.7
#>  [73] 0.7 0.7 0.7 0.7 0.7 0.7 0.3 0.7 0.7 0.7 0.7 0.7 0.7 0.3 0.7 0.7 0.7 0.7
#>  [91] 0.3 0.7 0.7 0.3 0.3 0.7 0.7 0.3 0.7 0.7

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

mean(D) # Expectation
#> [1] 0.7
median(D) # Median
#> [1] 1
mode(D) # Mode
#> [1] 1
var(D) # Variance
#> [1] 0.21
sd(D) # Standard Deviation
#> [1] 0.4582576
skew(D) # Skewness
#> [1] -0.8728716
kurt(D) # Excess Kurtosis
#> [1] -1.238095
entro(D) # Entropy
#> [1] 0.6108643
finf(D) # Fisher Information Matrix
#> [1] 4.761905

# List of all available moments
mom <- moments(D)
mom$mean # expectation
#> [1] 0.7

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

ll(D, x)
#> [1] -56.00264
llbern(x, p)
#> [1] -56.00264

ebern(x, type = "mle")
#> $prob
#> [1] 0.76
#> 
ebern(x, type = "me")
#> $prob
#> [1] 0.76
#> 

mle(D, x)
#> $prob
#> [1] 0.76
#> 
me(D, x)
#> $prob
#> [1] 0.76
#> 
e(D, x, type = "mle")
#> $prob
#> [1] 0.76
#> 

mle("bern", x) # the distr argument can be a character
#> $prob
#> [1] 0.76
#> 

# ------------------
# Estimator Variance
# ------------------

vbern(p, type = "mle")
#> prob 
#> 0.21 
vbern(p, type = "me")
#> prob 
#> 0.21 

avar_mle(D)
#> prob 
#> 0.21 
avar_me(D)
#> prob 
#> 0.21 

v(D, type = "mle")
#> prob 
#> 0.21

Usage

Arguments

Value

Details

See also

Examples