Distributacalcul (version 0.2.2)

MGF_negbinom: Moment Generating Function of the Negative Binomial distribution

Description

Moment Generating Function (PGF) of the Negative Binomial distribution with parameters \(r\) (number of successful trials) and \(p\) (probability of success).

Usage

MGF_negbinom(
  t,
  size,
  prob = (1/(1 + beta)),
  beta = ((1 - prob)/prob),
  nb_tries = FALSE
)

Arguments

t

t

size

Number of successful trials.

prob

Probability of success.

beta

Alternative parameterization of the negative binomial distribution where beta = (1 - p) / p.

nb_tries

logical; if FALSE (default) number of trials until the rth success, otherwise, number of failures until the rth success.

Value

Function :

• MGF_negbinom gives the moment generating function (MGF).

• PGF_negbinom gives the probability generating function (PGF).

• E_negbinom gives the expected value.

• V_negbinom gives the variance.

Invalid parameter values will return an error detailing which parameter is problematic.

Details

When \(k\) is the number of failures until the \(r\)th success, with a probability \(p\) of a success, the negative binomial has density: $$\left(\frac{r + k - 1}{k}\right) (p)^{r} (1 - p)^{k}$$ for \(k \in \{0, 1, \dots \}\)

When \(k\) is the number of trials until the \(r\)th success, with a probability \(p\) of a success, the negative binomial has density: $$\left(\frac{k - 1}{r - 1}\right) (p)^{r} (1 - p)^{k - r}$$ for \(k \in \{r, r + 1, r + 2, \dots \}\)

The alternative parameterization of the negative binomial with parameter \(\beta\), and \(k\) being the number of trials, has density: $$\frac{\Gamma(r + k)}{\Gamma(r) k!} \left(\frac{1}{1 + \beta}\right)^{r}% \left(\frac{\beta}{1 + \beta}\right)^{k - r}$$ for \(k \in \{0, 1, \dots \}\)

See Also

Other Negative Binomial Distribution: E_negbinom(), PGF_negbinom(), V_negbinom()

Examples

# NOT RUN {
MGF_negbinom(t = 1, size = 4, prob = 0.5)

# }