Common Distributions - Applications

Advanced distribution theory enables sophisticated modeling in science, engineering, and finance. We explore specialized distributions and their applications.

The Beta Distribution

Definition

$X \sim \text{Beta}(\alpha, \beta)$ on $[0,1]$ has PDF: $f(x) = \frac{1}{B(\alpha,\beta)} x^{\alpha-1}(1-x)^{\beta-1}, \quad 0 < x < 1$

where $B(\alpha,\beta) = \frac{\Gamma(\alpha)\Gamma(\beta)}{\Gamma(\alpha+\beta)}$ is the beta function.

Parameters: $\alpha, \beta > 0$ (shape parameters)

The beta distribution is extremely flexible for modeling proportions, probabilities, and percentages.

Special Cases:

Beta $(1,1) = \text{Uniform}(0,1)$
Beta $(\alpha, 1)$ concentrates near 1
Beta $(1, \beta)$ concentrates near 0
Beta $(\alpha, \beta)$ symmetric about 0.5 when $\alpha = \beta$

Example

Bayesian Analysis: Model uncertainty about a probability $p$ using Beta $(\alpha, \beta)$ prior.

If observing $k$ successes in $n$ trials (Binomial likelihood), posterior is: $p | \text{data} \sim \text{Beta}(\alpha + k, \beta + n - k)$

Starting with uniform prior Beta $(1,1)$ and observing 7 successes in 10 trials: $p | \text{data} \sim \text{Beta}(8, 4)$

Mean estimate: $\frac{8}{8+4} = 0.667$

The Weibull Distribution

Definition

$X \sim \text{Weibull}(k, \lambda)$ models lifetimes with PDF: $f(x) = \frac{k}{\lambda}\left(\frac{x}{\lambda}\right)^{k-1} e^{-(x/\lambda)^k}, \quad x \geq 0$

Parameters: $k > 0$ (shape), $\lambda > 0$ (scale)

Hazard Function: $h(x) = \frac{k}{\lambda}(x/\lambda)^{k-1}$

$k < 1$ : decreasing failure rate (infant mortality)
$k = 1$ : constant failure rate (Exponential $(\lambda)$ )
$k > 1$ : increasing failure rate (aging/wear-out)

Example

Wind speed modeling: $X \sim \text{Weibull}(2, 10)$ m/s (Rayleigh distribution).

Probability wind exceeds 15 m/s: $P(X > 15) = e^{-(15/10)^2} = e^{-2.25} \approx 0.105$

The t-Distribution

Definition

Student's $t$ -distribution with $\nu$ degrees of freedom: $f(x) = \frac{\Gamma((\nu+1)/2)}{\sqrt{\nu\pi}\Gamma(\nu/2)}\left(1 + \frac{x^2}{\nu}\right)^{-(\nu+1)/2}$

As $\nu \to \infty$ , $t_{\nu} \to \mathcal{N}(0,1)$ .

Usage: Inference about means when variance unknown and sample size small.

For sample mean $\bar{X}$ with sample std dev $S$ : $T = \frac{\bar{X} - \mu}{S/\sqrt{n}} \sim t_{n-1}$

Example

Sample of $n = 10$ has mean 52 and std dev 8. Test $H_0: \mu = 50$ vs. $H_1: \mu \neq 50$ .

$t = \frac{52 - 50}{8/\sqrt{10}} = \frac{2}{2.53} \approx 0.79$

With 9 degrees of freedom, $p$ -value $\approx 0.45$ . Do not reject $H_0$ .

The F-Distribution

Definition

If $U \sim \chi^2_m$ and $V \sim \chi^2_n$ independently: $F = \frac{U/m}{V/n} \sim F_{m,n}$

Parameters: $m, n$ (degrees of freedom)

Usage: Compare variances, ANOVA, regression F-tests.

Example

Testing equality of variances: $H_0: \sigma_1^2 = \sigma_2^2$ .

Sample variances $S_1^2 = 25$ (n=11), $S_2^2 = 16$ (n=16).

$F = \frac{S_1^2}{S_2^2} = \frac{25}{16} = 1.5625 \sim F_{10, 15}$

If critical value $F_{0.05}(10,15) \approx 2.54$ , do not reject $H_0$ .

Remark

These specialized distributions—Beta for proportions, Weibull for reliability, $t$ and $F$ for inference—extend the basic toolkit. Understanding when and how to apply each distribution is essential for advanced statistical modeling and requires both theoretical knowledge and practical experience.