Binom Distribution

The binomial distribution is a discrete probability distribution which models the probability of having x successes in n independent Bernoulli trials with success probability p.

Usage

Binom(size = 1, prob = 0.5)

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

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

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

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

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

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

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

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

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

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

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

llbinom(x, size, prob)

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

ebinom(x, size, type = "mle", ...)

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

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

vbinom(size, prob, type = "mle")

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

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

Arguments

size: number of trials (zero or more).
prob: numeric. Probability of success on each trial.
distr: an object of class Binom.
x: For the density function, x is a numeric vector of quantiles. For the moments functions, x is an object of class Binom. 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 mass function (PMF) of the binomial distribution is given by: $$ f(x; n, p) = \binom{n}{x} p^x (1 - p)^{n - x}, \quad N \in \mathbb{N}, \quad p \in (0, 1),$$ with $x \in \{0, 1, \dots, N\}$.

Examples

# -----------------------------------------------------
# Binomial Distribution Example
# -----------------------------------------------------

# Create the distribution
N <- 10 ; p <- 0.7
D <- Binom(N, p)

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

d(D, 0:N) # density function
#>  [1] 0.0000059049 0.0001377810 0.0014467005 0.0090016920 0.0367569090
#>  [6] 0.1029193452 0.2001209490 0.2668279320 0.2334744405 0.1210608210
#> [11] 0.0282475249
p(D, 0:N) # distribution function
#>  [1] 0.0000059049 0.0001436859 0.0015903864 0.0105920784 0.0473489874
#>  [6] 0.1502683326 0.3503892816 0.6172172136 0.8506916541 0.9717524751
#> [11] 1.0000000000
qn(D, c(0.4, 0.8)) # inverse distribution function
#> [1] 7 8
x <- r(D, 100) # random generator function

# alternative way to use the function
df <- d(D) ; df(x) # df is a function itself
#>   [1] 0.121060821 0.200120949 0.233474440 0.028247525 0.121060821 0.200120949
#>   [7] 0.266827932 0.233474440 0.266827932 0.200120949 0.200120949 0.102919345
#>  [13] 0.266827932 0.266827932 0.233474440 0.266827932 0.102919345 0.266827932
#>  [19] 0.200120949 0.266827932 0.200120949 0.266827932 0.102919345 0.200120949
#>  [25] 0.233474440 0.121060821 0.121060821 0.200120949 0.233474440 0.121060821
#>  [31] 0.036756909 0.266827932 0.121060821 0.266827932 0.121060821 0.036756909
#>  [37] 0.121060821 0.233474440 0.102919345 0.036756909 0.200120949 0.102919345
#>  [43] 0.036756909 0.121060821 0.266827932 0.266827932 0.009001692 0.233474440
#>  [49] 0.102919345 0.266827932 0.102919345 0.266827932 0.102919345 0.102919345
#>  [55] 0.233474440 0.028247525 0.233474440 0.102919345 0.233474440 0.102919345
#>  [61] 0.036756909 0.028247525 0.266827932 0.233474440 0.266827932 0.200120949
#>  [67] 0.102919345 0.233474440 0.102919345 0.200120949 0.200120949 0.121060821
#>  [73] 0.233474440 0.200120949 0.102919345 0.233474440 0.266827932 0.266827932
#>  [79] 0.266827932 0.233474440 0.028247525 0.233474440 0.266827932 0.200120949
#>  [85] 0.121060821 0.233474440 0.233474440 0.266827932 0.028247525 0.121060821
#>  [91] 0.009001692 0.102919345 0.121060821 0.121060821 0.233474440 0.266827932
#>  [97] 0.200120949 0.266827932 0.121060821 0.102919345

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

mean(D) # Expectation
#> [1] 7
var(D) # Variance
#> [1] 2.1
sd(D) # Standard Deviation
#> [1] 1.449138
skew(D) # Skewness
#> [1] -0.2760262
kurt(D) # Excess Kurtosis
#> [1] -0.1238095
entro(D) # Entropy
#> Warning: The entropy given is an approximation in the O(1 / n) order.
#> [1] 2.58229
finf(D) # Fisher Information Matrix
#> [1] 47.61905

# List of all available moments
mom <- moments(D)
#> Warning: The entropy given is an approximation in the O(1 / n) order.
mom$mean # expectation
#> [1] 7

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

ll(D, x)
#> [1] -193.9824
llbinom(x, N, p)
#> [1] -193.9824

ebinom(x, size = N, type = "mle")
#> $prob
#> [1] 0.694
#> 
ebinom(x, size = N, type = "me")
#> $prob
#> [1] 0.694
#> 

mle(D, x)
#> $prob
#> [1] 0.694
#> 
me(D, x)
#> $prob
#> [1] 0.694
#> 
e(D, x, type = "mle")
#> $prob
#> [1] 0.694
#> 

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

vbinom(N, p, type = "mle")
#>  prob 
#> 0.021 
vbinom(N, p, type = "me")
#>  prob 
#> 0.021 

avar_mle(D)
#>  prob 
#> 0.021 
avar_me(D)
#>  prob 
#> 0.021 

v(D, type = "mle")
#>  prob 
#> 0.021

Usage

Arguments

Value

Details

See also

Examples