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