Gamma & Beta Distributions

Two Distributions with Flexible Shapes

Gamma Distribution

$X \sim \text{Gamma}(\alpha, \beta)$ generalizes the exponential. It is the distribution of the waiting time for $\alpha$ events in a Poisson process with rate $\beta$:

$f(x) = \frac{\beta^\alpha}{\Gamma(\alpha)} x^{\alpha-1} e^{-\beta x}, \quad x > 0$

$E[X] = \frac{\alpha}{\beta}, \qquad \text{Var}(X) = \frac{\alpha}{\beta^2}$

Special case: $\text{Gamma}(1, \beta) = \text{Exponential}(\beta)$.

Beta Distribution

$X \sim \text{Beta}(\alpha, \beta)$ lives on $[0, 1]$ and models probabilities and proportions:

$f(x) = \frac{x^{\alpha-1}(1-x)^{\beta-1}}{B(\alpha,\beta)}, \quad 0 \leq x \leq 1$

$E[X] = \frac{\alpha}{\alpha+\beta}, \qquad \text{Var}(X) = \frac{\alpha\beta}{(\alpha+\beta)^2(\alpha+\beta+1)}$

Bayesian statistics: the Beta distribution is the conjugate prior for the Bernoulli/Binomial likelihood. After observing $k$ successes in $n$ trials with a $\text{Beta}(1,1)$ (uniform) prior, the posterior is $\text{Beta}(k+1, n-k+1)$.

# Beta(2, 5): modeling a conversion rate
alpha, beta = 2, 5
mean = alpha / (alpha + beta)
var  = alpha * beta / ((alpha + beta)**2 * (alpha + beta + 1))
print(round(mean, 4))  # 0.2857
print(round(var, 4))   # 0.0255

Your Task

Implement two functions:

• gamma_stats(alpha, beta) — prints $E[X]$ and $\text{Var}(X)$ of $\text{Gamma}(\alpha, \beta)$
• beta_stats(alpha, beta) — prints $E[X]$ and $\text{Var}(X)$ of $\text{Beta}(\alpha, \beta)$

Call both in sequence: first gamma_stats(2.0, 1.0), then beta_stats(2.0, 2.0).