mean_var_hp function

Mean and variance for hyper-Poisson distribution

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

mean_var_hp(mu, sigma)

mean_var_hp2(mu, sigma)

Arguments

• mu: value of the mu parameter.
• sigma: value of the sigma parameter.

Returns

the function returns a list with the mean and variance.

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.

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)

References

\insertRef saez2013hyperpoDiscreteDists

See Also

HYPERPO .

Author(s)

Freddy Hernandez, fhernanb@unal.edu.co