Math — Probability Distributions

A progressive study of four fundamental probability distributions implemented as Python classes — binomial, normal (Gaussian), Poisson, and exponential — with parameter estimation from data and PMF/PDF/CDF computation.

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Learning Objectives

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Task-by-Task Reference

Each task below highlights the unique challenge it posed and the new technique introduced — techniques from earlier tasks are not repeated.

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Task 0 — Binomial Distribution (binomial.py)

Challenge: Model the number of successes in $n$ independent Bernoulli trials, each with probability $p$ — implementing the binomial PMF from scratch using combinatorial formulas.

Approach: The constructor accepts either explicit $n$ and $p$ or estimates them from data. From data, compute the mean, then variance, then solve for $p = 1 - \sigma^2/\mu$ and $n = \text{round}(\mu/p)$. The PMF computes ${n \choose k} p^k (1-p)^{n-k}$ using iterative factorial accumulation to avoid overflow.

New techniques introduced:

Key takeaway: The binomial distribution models "number of successes in $n$ trials." Parameters can be estimated from data: $p = 1 - \text{variance}/\text{mean}$, then $n = \text{round}(\text{mean}/p)$.

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Task 1 — Normal Distribution (normal.py)

Challenge: Model the bell-shaped Gaussian distribution and compute probabilities on it — implementing the PDF formula $f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}$ from scratch.

Approach: Store $\mu$ (mean) and $\sigma$ (stddev) as floats. Provide z_score(x) to convert x-values to z-scores, x_value(z) to convert back, pdf(x) for the density, and cdf(x) using the error function approximation. Class constants e and pi are hardcoded for precision control.

New techniques introduced:

Key takeaway: The normal distribution is defined by $\mu$ (center) and $\sigma$ (spread). Z-scores standardize any normal to $\mathcal{N}(0,1)$. The CDF answers "what's the probability of being below x?"

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Task 2 — Poisson Distribution (poisson.py)

Challenge: Model the number of events occurring in a fixed interval — implementing the Poisson PMF $P(k) = \frac{\lambda^k e^{-\lambda}}{k!}$ and CDF as a sum.

Approach: The rate parameter $\lambda$ (lambtha) is either given or estimated as the sample mean. The PMF computes $\lambda^k e^{-\lambda} / k!$ using iterative factorial accumulation. The CDF sums PMF values from $0$ to $k$ using the same iterative factorial approach for efficiency.

New techniques introduced:

Key takeaway: The Poisson distribution models count data — "how many events in a fixed interval?" $\lambda$ is both the mean AND the variance. The PMF uses $e^{-\lambda}$ as the base probability of zero events.

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Task 3 — Exponential Distribution (exponential.py)

Challenge: Model the waiting time between events in a Poisson process — implementing the exponential PDF $f(x) = \lambda e^{-\lambda x}$ and CDF $F(x) = 1 - e^{-\lambda x}$.

Approach: The rate $\lambda$ is either given or estimated as $1/\text{mean}$ of the data (the reciprocal of the sample mean). The PDF computes $\lambda e^{-\lambda x}$ directly. The CDF uses $1 - e^{-\lambda x}$ — a simple closed form, unlike the Poisson which requires summation.

New techniques introduced:

Key takeaway: The exponential distribution is the continuous counterpart to the discrete Poisson. It models waiting times with the "memoryless" property: $P(X > s+t \mid X > s) = P(X > t)$. The rate $\lambda$ is the inverse of the expected waiting time.

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Technique Inventory

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Resources

• SciPy Stats — Probability Distributions
• Binomial Distribution — Wikipedia
• Normal Distribution — Wikipedia
• Poisson Distribution — Wikipedia
• Exponential Distribution — Wikipedia