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 nnn independent Bernoulli trials, each with probability ppp — implementing the binomial PMF from scratch using combinatorial formulas.

Approach: The constructor accepts either explicit nnn and ppp or estimates them from data. From data, compute the mean, then variance, then solve for p=1−σ2/μp = 1 - \sigma^2/\mup=1−σ2/μ and n=round(μ/p)n = \text{round}(\mu/p)n=round(μ/p). The PMF computes (nk)pk(1−p)n−k{n \choose k} p^k (1-p)^{n-k}(kn​)pk(1−p)n−k using iterative factorial accumulation to avoid overflow.

New techniques introduced:

Key takeaway: The binomial distribution models "number of successes in nnn trials." Parameters can be estimated from data: p=1−variance/meanp = 1 - \text{variance}/\text{mean}p=1−variance/mean, then n=round(mean/p)n = \text{round}(\text{mean}/p)n=round(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)=1σ2πe−(x−μ)22σ2f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}f(x)=σ2π​1​e−2σ2(x−μ)2​ from scratch.

Approach: Store μ\muμ (mean) and σ\sigmaσ (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 μ\muμ (center) and σ\sigmaσ (spread). Z-scores standardize any normal to N(0,1)\mathcal{N}(0,1)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)=λke−λk!P(k) = \frac{\lambda^k e^{-\lambda}}{k!}P(k)=k!λke−λ​ and CDF as a sum.

Approach: The rate parameter λ\lambdaλ (lambtha) is either given or estimated as the sample mean. The PMF computes λke−λ/k!\lambda^k e^{-\lambda} / k!λke−λ/k! using iterative factorial accumulation. The CDF sums PMF values from 000 to kkk 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?" λ\lambdaλ is both the mean AND the variance. The PMF uses e−λe^{-\lambda}e−λ 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)=λe−λxf(x) = \lambda e^{-\lambda x}f(x)=λe−λx and CDF F(x)=1−e−λxF(x) = 1 - e^{-\lambda x}F(x)=1−e−λx.

Approach: The rate λ\lambdaλ is either given or estimated as 1/mean1/\text{mean}1/mean of the data (the reciprocal of the sample mean). The PDF computes λe−λx\lambda e^{-\lambda x}λe−λx directly. The CDF uses 1−e−λx1 - e^{-\lambda x}1−e−λx — 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∣X>s)=P(X>t)P(X > s+t \mid X > s) = P(X > t)P(X>s+t∣X>s)=P(X>t). The rate λ\lambdaλ 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