Random Variables & Distributions

The problem, from first principles

Random variables

Every ML model that outputs a probability, a classifier's softmax, a language model's next-token distribution, a VAE's latent code, is treating some quantity as a random variable: a variable whose value isn't fixed, but drawn from a distribution. A discrete random variable takes one of a countable set of values (a class label, a token id, a die roll); its distribution is a probability mass function (PMF), a table of P(X = x) for each possible x. A continuous random variable takes any value in a range (a pixel intensity, a sensor reading); it doesn't have a PMF at all, a single exact value has probability zero, instead it has a probability density function (PDF), covered in the next question (01-random-variables). This question stays entirely in the discrete case and nails down the two things every valid PMF must satisfy, plus the single most useful summary of a random variable: its expected value.

PMFs & PDFs

01-random-variables introduced the PMF for a discrete random variable: a table of P(X = x) for each possible value. But most real quantities a model touches, a pixel value, an activation, a continuous latent variable, aren't discrete and a PMF literally cannot describe them: for a continuous variable, the probability of hitting any exact real number is zero (there are infinitely many of them to spread the probability over). A probability density function (PDF) solves this by describing probability as area under a curve instead of height at a point, P(a <= X <= b) is the integral of the density between a and b, not a single lookup. This question implements one canonical example of each: the discrete Binomial PMF (counting successes over n trials) and the continuous Uniform PDF (equally likely anywhere in an interval).

From theory to code

Random variables

Implement is_valid_pmf(probabilities), checking the two axioms every PMF must satisfy, then expected_value_discrete(outcomes, probabilities), computing E[X]. The signatures and docstrings are already in the editor.

PMFs & PDFs

Implement binomial_pmf(n, p, k), returning P(X = k) for a Binomial(n, p) random variable, then uniform_pdf(x, a, b), returning the constant density of a Uniform(a, b) random variable at point x. The signatures and docstrings are already in the editor.

Constraints

Random variables

• probabilities is a list/tuple of floats.
• outcomes and probabilities are the same length, and probabilities[i] is P(X = outcomes[i]).
• "Sums to 1" means within 1e-9 of exactly 1, to tolerate floating-point roundoff.

PMFs & PDFs

• n is a positive integer, k is an integer in [0, n], p is a float in [0, 1].
• a < b for uniform_pdf; x can be any real number, including outside [a, b].

Hints

Random variables

PMFs & PDFs