Random Variables & Distributions: Models for How Data Arrive

Connect Bernoulli, binomial, Poisson and continuous distributions to real data-generating processes and ML targets.

A distribution is not merely a curve. It is a model for how values are generated.

Choosing a distribution begins with the mechanism:

Is the outcome binary? A count? A waiting time? A continuous measurement?

TipWhat you will learn

• Random variables, PMFs/PDFs and CDFs
• Expectation and variance
• Bernoulli, binomial and Poisson models
• How parameters change distributions
• How distribution choice appears in ML loss functions

Random variable

A random variable maps uncertain outcomes to numbers.

Its expectation is

E[X]=\sum_x xP(X=x)

for a discrete variable, or

E[X]=\int x f(x)\,dx

for a continuous variable.

Variance is

\operatorname{Var}(X)=E[(X-E[X])^2].

Interactive distribution laboratory

Choose a probability p and see the binomial count distribution change.

viewof nbin = Inputs.range([5,100], {value:20, step:1, label:"Trials n"})
viewof pbin = Inputs.range([0.02,.98], {value:.3, step:.01, label:"Success probability p"})
logFact = n => d3.sum(d3.range(1,n+1), d=>Math.log(d))
choose = (n,k)=>Math.exp(logFact(n)-logFact(k)-logFact(n-k))
binom = d3.range(nbin+1).map(k=>({k,p:choose(nbin,k)*pbin**k*(1-pbin)**(nbin-k)}))
Plot.plot({
 height:320,x:{label:"Number of successes"},y:{label:"Probability"},
 marks:[Plot.barY(binom,{x:"k",y:"p",fill:"#4F46E5"}),Plot.ruleY([0])]
})

For X\sim\text{Binomial}(n,p),

E[X]=np,\qquad \operatorname{Var}(X)=np(1-p).

Mechanism → distribution

Data-generating story	Common starting model
One yes/no outcome	Bernoulli
Successes in fixed n trials	Binomial
Events in an interval	Poisson
Symmetric continuous measurement	Normal
Positive waiting time	Exponential/Gamma

The model is an approximation. The story helps you choose a sensible starting point.

Simulation: Poisson counts

import numpy as np
import plotly.express as px
import pandas as pd

rng = np.random.default_rng(4)
rows=[]
for lam in [1,4,10]:
    for x in rng.poisson(lam, 3000):
        rows.append({"lambda":f"λ={lam}","count":x})
df=pd.DataFrame(rows)
fig=px.histogram(df,x="count",facet_col="lambda",histnorm="probability",
                 nbins=25)
fig.update_layout(height=340,showlegend=False)
fig.show()

DS/ML Bridge: distributions hide inside loss functions

A statistical model often assumes a conditional distribution:

Regression with squared error
      ↕
Gaussian noise assumption

Binary cross-entropy
      ↕
Bernoulli likelihood

Poisson regression
      ↕
Poisson count likelihood

This is why statistics and machine learning are deeply connected: many losses are negative log-likelihoods under probabilistic models.

NoteCompanion exploration

Your interactive Normal Distribution article is the natural deep dive for the Gaussian case: use it to explore \mu, \sigma, the 68–95–99.7 rule, bivariate normal geometry and Q–Q plots.

Summary

• A distribution describes a data-generating mechanism probabilistically.
• Expectation describes long-run center; variance describes dispersion.
• Bernoulli/binomial are natural for binary trials; Poisson for event counts.
• Distribution assumptions reappear as ML objectives and likelihoods.

Next: Sampling, Standard Error & the Central Limit Theorem