11.3 Gamma and beta distributions

The gamma distribution models continuous, positive quantities like waiting times or failure rates. It's defined by shape and rate parameters, which determine its form and scale. Understanding its probability density function and cumulative distribution function is crucial for calculating probabilities and quantiles.

The beta distribution is perfect for modeling proportions and probabilities between 0 and 1. Its two shape parameters control its appearance, making it versatile for various scenarios. It's especially useful in Bayesian inference, where parameters can be interpreted as pseudo-counts of successes and failures.

Gamma Distribution

Gamma distribution and parameters

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• Continuous probability distribution models waiting times, time until failure, or other positive, continuous quantities
• Defined by two parameters:
  • Shape parameter $\alpha > 0$ determines the shape of the distribution (exponential, bell-shaped, or skewed)
  • Rate parameter $\beta > 0$ controls the scale and rate of decay
• Probability density function (PDF) for a gamma-distributed random variable $X$, where $x > 0$: $f(x; \alpha, \beta) = \frac{\beta^\alpha}{\Gamma(\alpha)} x^{\alpha-1} e^{-\beta x}$
  • $\Gamma(\alpha)$ represents the gamma function, an extension of the factorial function to real and complex numbers, defined as $\Gamma(\alpha) = \int_0^\infty x^{\alpha-1} e^{-x} dx$
• Mean of the gamma distribution equals $\frac{\alpha}{\beta}$, while the variance is $\frac{\alpha}{\beta^2}$

Probabilities in gamma distributions

• Calculate probabilities for a gamma-distributed random variable $X$ using the cumulative distribution function (CDF)
  • CDF $F(x; \alpha, \beta)$ represents the probability that $X$ is less than or equal to a specific value $x$ and is defined as $F(x; \alpha, \beta) = \int_0^x f(t; \alpha, \beta) dt$
  • Probability $P(X \leq x)$ equals the CDF evaluated at $x$, i.e., $F(x; \alpha, \beta)$
• Determine quantiles by inverting the CDF to find the value $x_p$ that corresponds to a given probability $p$
  • The $p$-th quantile $x_p$ satisfies the equation $F(x_p; \alpha, \beta) = p$
  • Quantiles help establish probability intervals and percentiles (median, quartiles)

Gamma vs exponential distributions

• Exponential distribution is a special case of the gamma distribution when the shape parameter $\alpha = 1$
  • Exponential PDF simplifies to $f(x; \beta) = \beta e^{-\beta x}$ for $x > 0$, where $\beta$ is the rate parameter
• Sum of $n$ independent exponentially distributed random variables with rate $\beta$ follows a gamma distribution
  • Shape parameter becomes $\alpha = n$, while the rate parameter $\beta$ remains unchanged
  • Relationship is useful for modeling total waiting times (customer service) or system reliability (time until $n$ components fail)

Beta Distribution

Beta distribution for proportions

• Continuous probability distribution defined on the interval $[0, 1]$, making it suitable for modeling proportions, probabilities, and fractions
• Characterized by two shape parameters $\alpha > 0$ and $\beta > 0$, which determine the shape of the distribution (symmetric, skewed, U-shaped, or J-shaped)
• PDF of a beta-distributed random variable $X$, where $0 < x < 1$: $f(x; \alpha, \beta) = \frac{1}{B(\alpha, \beta)} x^{\alpha-1} (1-x)^{\beta-1}$
  • $B(\alpha, \beta)$ is the beta function, a normalization constant ensuring the PDF integrates to 1, defined as $B(\alpha, \beta) = \int_0^1 x^{\alpha-1} (1-x)^{\beta-1} dx$
• Mean of the beta distribution is $\frac{\alpha}{\alpha + \beta}$, and the variance is $\frac{\alpha \beta}{(\alpha + \beta)^2 (\alpha + \beta + 1)}$
• Applications include modeling the proportion of defective items in a batch (quality control) or the success probability of a binary event (coin flips, survey responses)
• Shape parameters $\alpha$ and $\beta$ can be interpreted as pseudo-counts in Bayesian inference
  • $\alpha$ represents the number of successes plus one, while $\beta$ represents the number of failures plus one
  • Interpretation allows for updating prior beliefs about a proportion based on observed data (posterior distribution)