Geometric Distribution

The Geometric distribution is a discrete probability distribution that models the number of failures before the first success in a sequence of independent Bernoulli trials, each with success probability $0 < p \leq 1$.

Usage

Geom(prob = 0.5)

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

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

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

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

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

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

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

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

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

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

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

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

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

llgeom(x, prob)

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

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

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

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

vgeom(prob, type = "mle")

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

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

Arguments

prob: numeric. Probability of success.
distr: an object of class Geom.
x: For the density function, x is a numeric vector of quantiles. For the moments functions, x is an object of class Geom. 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 Geometric distribution is: $$ P(X = k) = (1 - p)^k p, \quad k \in \mathbb{N}_0.$$

Examples

# -----------------------------------------------------
# Geom Distribution Example
# -----------------------------------------------------

# Create the distribution
p <- 0.4
D <- Geom(p)

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

d(D, 0:4) # density function
#> [1] 0.40000 0.24000 0.14400 0.08640 0.05184
p(D, 0:4) # distribution function
#> [1] 0.40000 0.64000 0.78400 0.87040 0.92224
qn(D, c(0.4, 0.8)) # inverse distribution function
#> [1] 0 3
x <- r(D, 100) # random generator function

# alternative way to use the function
df <- d(D) ; df(x) # df is a function itself
#>   [1] 0.03110400 0.40000000 0.24000000 0.24000000 0.05184000 0.24000000
#>   [7] 0.24000000 0.40000000 0.14400000 0.24000000 0.40000000 0.40000000
#>  [13] 0.40000000 0.01866240 0.03110400 0.08640000 0.40000000 0.40000000
#>  [19] 0.40000000 0.40000000 0.14400000 0.24000000 0.40000000 0.05184000
#>  [25] 0.05184000 0.40000000 0.14400000 0.24000000 0.40000000 0.40000000
#>  [31] 0.40000000 0.14400000 0.24000000 0.05184000 0.01866240 0.14400000
#>  [37] 0.24000000 0.40000000 0.08640000 0.24000000 0.40000000 0.14400000
#>  [43] 0.40000000 0.24000000 0.01119744 0.40000000 0.40000000 0.05184000
#>  [49] 0.14400000 0.40000000 0.40000000 0.40000000 0.40000000 0.40000000
#>  [55] 0.14400000 0.24000000 0.40000000 0.40000000 0.24000000 0.08640000
#>  [61] 0.40000000 0.40000000 0.40000000 0.14400000 0.40000000 0.40000000
#>  [67] 0.40000000 0.08640000 0.08640000 0.14400000 0.05184000 0.24000000
#>  [73] 0.40000000 0.40000000 0.40000000 0.08640000 0.14400000 0.40000000
#>  [79] 0.24000000 0.24000000 0.40000000 0.24000000 0.01866240 0.40000000
#>  [85] 0.14400000 0.14400000 0.24000000 0.08640000 0.05184000 0.24000000
#>  [91] 0.05184000 0.14400000 0.40000000 0.40000000 0.24000000 0.40000000
#>  [97] 0.40000000 0.40000000 0.05184000 0.08640000

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

mean(D) # Expectation
#> [1] 1.5
median(D) # Median
#> [1] 1
mode(D) # Mode
#> [1] 0
var(D) # Variance
#> [1] 3.75
sd(D) # Standard Deviation
#> [1] 1.936492
skew(D) # Skewness
#> [1] 2.065591
kurt(D) # Excess Kurtosis
#> [1] 6.266667
entro(D) # Entropy
#> [1] 1.682529
finf(D) # Fisher Information Matrix
#> [1] 10.41667

# List of all available moments
mom <- moments(D)
mom$mean # expectation
#> [1] 1.5

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

ll(D, x)
#> [1] -164.6771
llgeom(x, p)
#> [1] -164.6771

egeom(x, type = "mle")
#> $prob
#> [1] 0.4115226
#> 
egeom(x, type = "me")
#> $prob
#> [1] 0.4115226
#> 

mle(D, x)
#> $prob
#> [1] 0.4115226
#> 
me(D, x)
#> $prob
#> [1] 0.4115226
#> 
e(D, x, type = "mle")
#> $prob
#> [1] 0.4115226
#> 

mle("geom", x) # the distr argument can be a character
#> $prob
#> [1] 0.4115226
#> 

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

vgeom(p, type = "mle")
#>  prob 
#> 0.096 
vgeom(p, type = "me")
#>  prob 
#> 0.096 

avar_mle(D)
#>  prob 
#> 0.096 
avar_me(D)
#>  prob 
#> 0.096 

v(D, type = "mle")
#>  prob 
#> 0.096

Usage

Arguments

Value

Details

See also

Examples