The degenerate distribution

The degenerate distribution takes a single value which is certain to be observed. It takes a single parameter, which is the value that is observed by the distribution.

dist_degenerate(x)

Arguments

x

The value of the distribution.

Details

We recommend reading this documentation on https://pkg.mitchelloharawild.com/distributional/, where the math will render nicely.

In the following, let \(X\) be a degenerate random variable with value x = \(k_0\).

Support: \(R\), the set of all real numbers

Mean: \(k_0\)

Variance: \(0\)

Probability density function (p.d.f):

$$ f(x) = 1 for x = k_0 $$ $$ f(x) = 0 for x \neq k_0 $$

Cumulative distribution function (c.d.f):

The cumulative distribution function has the form

$$ F(x) = 0 for x < k_0 $$ $$ F(x) = 1 for x \ge k_0 $$

Moment generating function (m.g.f):

$$ E(e^{tX}) = e^{k_0 t} $$

Examples

dist_degenerate(x = 1:5)
#> <distribution[5]>
#> [1] 1 2 3 4 5