Here, we provide the PDFs for several distributions that we use throughout the book.

FormalPara Beta Distribution

The probability of observing the random variable x under the Beta distribution with shape parameters α ∈ (0, ∞) and β ∈ (0, ∞) is

$$\displaystyle \begin{aligned} f(x|\alpha,\beta) = \dfrac{\varGamma(\alpha)\varGamma(\beta)}{\varGamma(\alpha+\beta)} x^{\alpha-1} (1-x)^{\beta-1} \end{aligned}$$

where Γ(x) = (x − 1)!.

FormalPara Binomial Distribution

In n trials, the binomial distribution defines the probability of observing x = {0, 1, …, n} successes as

$$\displaystyle \begin{aligned} f(x|p,n) = {n \choose x} p^{x} (1-p)^{n-x}, \end{aligned}$$

where the probability of a single-trial success is the parameter \(p\in \left [ 0,1 \right ]\).

FormalPara Gamma Distribution

The probability of observing the random variable x under the Gamma distribution with shape parameter k ∈ (0, ∞) and scale parameter θ ∈ (0, ∞) is

$$\displaystyle \begin{aligned} f(x|k,\theta) = \dfrac{1}{\varGamma(k) \theta^k} x^{k-1} \exp \left(-\dfrac{x}{\theta} \right). \end{aligned}$$