11.5. Summary#

Below is a summary table of the four previously discussed common probability distributions.

Name

PMF or PDF

Mean

Variance

Type

Parameter(s)

Bernoulli

\(\begin{cases} p & x=1\\ 1-p & x=0 \end{cases}\)

\(\sum_x x p(x)\)

\(\sigma_X^2 = \sum_x (x - \mu)^2 p(x)\)

Discrete

\(p\)

Binomial

\(\begin{cases} \frac{n!}{x!(n-x)!} p^x (1-p)^{n-x} & x=0,1,...,n\\ 0 & \mathrm{otherwise} \end{cases}\)

\(n p\)

\((1-p)\)

Discrete

\(p\), \(n\), \(x\)

Poisson

\(\begin{cases} e^{-\lambda} \frac{\lambda^x}{x!} & \text{if } x \text{ is a non-negative integer} \\ 0 & \mathrm{otherwise} \end{cases}\)

\(\lambda\)

\(\lambda\)

Discrete

\(\lambda\)

Normal

\(f(x) = \frac{1}{\sigma \sqrt{2 \pi}} e^{-(x-\mu)^2 / (2\sigma^2)}\)

\(\mu\)

\(\sigma^2\)

Continuous

\(\mu\), \(\sigma\)