Common Distributions - Examples and Constructions

Real-world applications demonstrate how to select appropriate distributions based on the underlying data-generating process.

Modeling Count Data

Quality Control: A factory inspects batches of 100 items. The number of defective items follows Binomial$(100, p)$ where $p$ is the defect rate.

If $p = 0.02$: $P(\text{at most 3 defects}) = \sum_{k=0}^3 \binom{100}{k} (0.02)^k (0.98)^{100-k}$

For rare defects ($p$ small), Poisson$(np = 2)$ approximation: $P(X \leq 3) = \sum_{k=0}^3 \frac{e^{-2} \cdot 2^k}{k!} \approx 0.857$

Modeling Continuous Measurements

Measurement Errors: Scientific measurements often follow normal distributions due to the aggregation of many small independent errors (CLT).

Heights in a population: $X \sim \mathcal{N}(170, 100)$ cm (mean 170cm, std dev 10cm).

Probability someone is taller than 185cm: $P(X > 185) = 1 - \Phi\left(\frac{185-170}{10}\right) = 1 - \Phi(1.5) \approx 0.0668$

About 6.7% are taller than 185cm.

Reliability and Survival Analysis

Component Lifetimes: Electronic components often have exponentially distributed lifetimes if failures occur at constant rate.

For a component with mean lifetime 5 years (Exponential$(1/5)$): $P(\text{lasts } > 10 \text{ years}) = e^{-10/5} = e^{-2} \approx 0.135$

System Reliability: For a system with $n$ independent components, each with lifetime Exponential$(\lambda)$:

• Series (all must work): $T_{\text{min}} = \min\{T_1, \ldots, T_n\} \sim \text{Exponential}(n\lambda)$
• Parallel (one must work): $T_{\text{max}} = \max\{T_1, \ldots, T_n\}$ has more complex distribution

Financial Applications

Stock Returns: Daily log-returns often modeled as normal. For a stock with annual return 8% and volatility 20%:

Daily return $\sim \mathcal{N}(0.08/252, (0.20)^2/252)$ (252 trading days/year)

Value at Risk (VaR): For a portfolio with value V \sim \mathcal{N}(\1M, $100K^2)$:

95% VaR (loss exceeded 5% of time): $\text{VaR}_{0.05} = \mu - 1.645\sigma = 1M - 1.645(100K) \approx \$835,500$

Expected loss is at most about $165K with 95% confidence.

Queueing Theory

Bank Teller: Customers arrive at rate $\lambda = 10$/hour (Poisson process). Service time per customer is Exponential$(12/\text{hour})$ (mean 5 minutes).

Number of arrivals in one hour: Poisson$(10)$

Time until first arrival: Exponential$(10)$ with mean 6 minutes

For stability, arrival rate must be less than service rate: $\lambda < \mu$ (here $10 < 12$ ✓)