DiscreteDists (version 1.0.0)

mean_var_hp: Mean and variance for hyper-Poisson distribution

Description

This function calculates the mean and variance for the hyper-Poisson distribution with parameters \(\mu\) and \(\sigma\).

Usage

mean_var_hp(mu, sigma)
mean_var_hp2(mu, sigma)

Value

the function returns a list with the mean and variance.

Arguments

mu

value of the mu parameter.

sigma

value of the sigma parameter.

Author

Freddy Hernandez,

Details

The hyper-Poisson distribution with parameters \(\mu\) and \(\sigma\) has a support 0, 1, 2, ... and density given by

\(f(x | \mu, \sigma) = \frac{\mu^x}{_1F_1(1;\mu;\sigma)}\frac{\Gamma(\sigma)}{\Gamma(x+\sigma)}\)

where the function \(_1F_1(a;c;z)\) is defined as

\(_1F_1(a;c;z) = \sum_{r=0}^{\infty}\frac{(a)_r}{(c)_r}\frac{z^r}{r!}\)

and \((a)_r = \frac{\gamma(a+r)}{\gamma(a)}\) for \(a>0\) and \(r\) positive integer.

This function calculates the mean and variance of this distribution.

References

saez2013hyperpoDiscreteDists

See Also

HYPERPO.

Examples

# Example 1

# Theoretical values
mean_var_hp(mu=5.5, sigma=0.1)

# Using simulated values
y <- rHYPERPO(n=1000, mu=5.5, sigma=0.1)
mean(y)
var(y)


# Example 2

# Theoretical values
mean_var_hp2(mu=5.5, sigma=1.9)

# Using simulated values
y <- rHYPERPO2(n=1000, mu=5.5, sigma=1.9)
mean(y)
var(y)