Probability and Statistics Lab

Discrete

Bernoulli distribution

A single trial with two outcomes: success with probability p and failure with probability 1-p.

\(Bernoulli(p)\)

Discrete

Binomial distribution

Counts successes in n independent Bernoulli trials with common success probability p.

\(Binomial(n,p)\)

Discrete

Poisson distribution

Counts events occurring independently at a constant average rate \(\lambda\) over a fixed interval.

\(Poisson(\lambda)\)

Discrete

Hypergeometric distribution

Counts successes in K draws without replacement from a population of size N containing M successes.

\(Hypergeometric(N,M,K)\)

Discrete

Geometric distribution

Models the trial number on which the first success occurs.

\(Geometric(p)\)

Discrete

Negative binomial distribution

Models the trial number on which the r-th success occurs.

\(Negative Binomial(r,p)\)

Discrete

Discrete uniform distribution

Assigns equal probability to every integer between a and b inclusive.

\(Discrete Uniform(a,b)\)

Continuous

Uniform distribution

Chooses a point from an interval so probability is proportional to interval length.

\(Uniform(a,b)\)

Continuous

Triangular distribution

A bounded continuous distribution with lower bound a, mode c, and upper bound b.

\(Triangular(a,c,b)\)

Continuous

Exponential distribution

A waiting-time distribution for events occurring independently at a constant rate.

\(Exponential(\lambda)\)

Continuous

Normal distribution

The classic bell curve, symmetric around \(\mu\) with variance \(\sigma^2\).

\(Normal(\mu,\sigma^2)\)

Continuous

Chi-square distribution

A sampling distribution formed by summing squared independent standard normal variables.

\(\chi^2(\nu)\)

Continuous

F distribution

The ratio of two independent chi-square variables scaled by their degrees of freedom.

\(F(\nu_1,\nu_2)\)

Confidence intervals

Visualize the sampling uncertainty around the estimate while computing the interval.

Hypothesis testing

See the rejection region, test statistic, p-value, and decision together.

Lookup tables and power

Connect critical values and tail areas to the power calculation.

Statistical tables

Use the Z, t, chi-square, and F tables in the same style as the notes.

Finite population sampling

Rank random numbers and return a sample without replacement.

Bayes theorem simulation

Simulate the red-red, white-white, and mixed-card question.

Random number generators

Generate and inspect a linear congruential sequence.

German tank problem

Sample plate numbers and estimate the hidden population size.

Galton board

Drop animated balls through pegs with a chosen right-move probability.

Poisson approximation

Choose n and p, then compare the binomial PMF with Poisson(lambda=np).

Birthday paradox

Add people to a calendar heat map and see shared birthdays appear.

Disclosure of ballot box results

Reveal ballot boxes over time and track intervals and a one-sided test.

Monty Hall

Choose the number of doors and compare staying with switching.

Airplane seat problem

Let one passenger lose a ticket, then watch whether the last passenger gets the assigned seat.