We recommend reading this documentation on
https://alexpghayes.github.io/distributions3/, where the math
will render with additional detail and much greater clarity.
In the following, let \(X\) be an Exponential random variable with
rate parameter rate
= \(\lambda\).
Support: \(x \in (0, \infty)\)
Mean: \(\frac{1}{\lambda}\)
Variance: \(\frac{1}{\lambda^2}\)
Probability density function (p.d.f):
$$
f(x) = \lambda e^{-\lambda x}
$$
Cumulative distribution function (c.d.f):
$$
F(x) = 1 - e^{-\lambda x}
$$
Moment generating function (m.g.f):
$$
\frac{\lambda}{\lambda - t}, for t < \lambda
$$