Random variables, distributions, and transformations

Probability models assign masses to events; random variables map their outcomes to values. This lesson derives discrete distributions, separates density from probability, constructs distribution functions and quantiles, and transforms variables without losing preimages or Jacobian factors. Each model is attached to a generating story and explicit support, rather than selected because a curve looks familiar.

Retrieval check: Module 21 supplies event models, conditioning and independence. Continuous AI branch: Module 17 is required for densities, continuous sampling derivations and changes of variables. The CS route follows all discrete material, uses the discrete bridge in Section 3, and omits the labelled continuous portions of labs and exercises. Those minutes can be spent on discrete CDF and transformation practice within the same schedule.

A real random variable X maps Ω to R. In a finite model any such map is valid; in general measurability requires {ω:X(ω)≤t} to be an event for every real t. The random variable is the whole function before the experiment occurs. A realised value x=X(ω) is an observation, and an outcome ω can contain more information than x. For two coin outcomes, mapping to the number of heads merges HT and TH into the same numerical value one. Outcomes, variables and observations are therefore distinct objects.

The distribution of X assigns probability to its value sets through preimages: P(X∈S)=P({ω:X(ω)∈S}). Distinct maps or experiments can have the same distribution. Conversely the same displayed values can have different masses in different models. For a discrete variable, its PMF p_X(x)=P(X=x) is nonnegative and sums to one over its countable support. The support in this discrete sense consists of positive-mass values; values outside it have mass zero. For a continuous distribution we will describe supported intervals or neighbourhoods, since individual point masses are zero.

The cumulative distribution function F_X(t)=P(X≤t) exists for discrete, continuous and mixed variables. It is nondecreasing because the events increase with t; limits at −∞ and +∞ are zero and one by probability continuity on nested events. It is right-continuous: if t_k↓t, then {X≤t_k} decreases to {X≤t}, and Module 21’s continuity property gives F(t_k)→F(t). Its left limit F(t−)=P(X<t) follows from increasing thresholds approaching t from below. Therefore its jump at t is exactly P(X=t).

For any real a<b, P(a<X≤b)=F(b)−F(a). Both left strictness and right inclusion matter at atoms. P(a≤X≤b)=F(b)−F(a−), while P(X>b)=1−F(b) and P(X≥b)=1−F(b−). These identities follow by disjoint event decompositions, not by assuming a density. In a continuous distribution with no endpoint masses some distinctions coincide numerically; a discrete model cannot silently discard them. An empirical CDF is also a step function with jumps at observations, even when the generating distribution is continuous.

For a finite table, calculate F by cumulative masses in sorted value order and retain the convention ≤. If a graph connects CDF jump endpoints by sloping lines, it falsely suggests intermediate probability changes where the discrete CDF is constant. PMF bars and CDF steps communicate different quantities: one is probability at a value, the other probability up to a threshold. Axes should label which object is being shown, and transformed values must be sorted again after their masses have been combined.

The distribution alone does not specify joint dependence with another variable. Two variables can both be fair Bernoulli while being equal, opposite or independent. Their marginal PMFs match in all three cases, yet joint events differ. The next module builds joint distributions and expectations. For now, avoid multiplying marginal masses without the independence assumptions from Module 21. A function of an existing variable also retains dependence on that variable, even if its resulting distribution has a familiar name.

A distribution is also different from its representation precision. Rounding a continuous measurement into bins creates a discrete reported variable whose mass in each bin is the original interval probability. If a sensor rounds to tenths, an observed label .3 can represent a whole interval of underlying inputs rather than the single real value .3. State the measurement and tie convention before comparing exact point observations with density models. This explains why repeated rounded labels do not, by themselves, prove the underlying physical quantity has atoms.

A Bernoulli variable takes values0,1 with probabilities1−p,p, where0≤p≤1. It often indicates whether an event occurred. The endpoint models p=0 and p=1 are valid degenerate distributions. A categorical variable takes one of k labels with masses p_i≥0 summing to one; coding labels as integers does not make their numerical distances meaningful. A sampled class index is categorical, whereas a vector of predicted class probabilities describes a distribution rather than a realised class.

A binomial variable counts successes in a fixed number n of independent Bernoulli trials with the same success probability p. For k=0,…,n, choose which k trials succeed in comb(n,k) ways; every such pattern has mass p^k(1−p)^{n−k}, so p_X(k)=comb(n,k)p^k(1−p)^{n−k}. The disjoint success-count events partition all trial patterns, or the binomial theorem normalises the sum. Handle p=0,1 directly rather than rely on an ambiguous numerical 0^0. Independence, equal p and fixed n are part of the model, not consequences of observing a count.

The geometric trial count T is the index of the first success in independent repeated Bernoulli trials with a fixed0<p≤1. Its support starts at1 and P(T=k)=(1−p)^{k−1}p. A failures-before-success convention uses G=T−1 with support starting at0 and P(G=k)=(1−p)^k p. Always state which one is used. For0<p<1, the first K trial masses sum to1−(1−p)^K and the remaining tail is(1−p)^K. When p=1, T=1; when p=0 there is no finite first success, so this finite-valued geometric PMF is not normalised.

The geometric tail P(T>m)=(1−p)^m for integers m≥0 gives memorylessness: P(T>m+n|T>m)=(1−p)^n when the conditioning mass is positive. Surviving m failures does not alter the waiting mechanism under independent identical trials. A system whose success rate changes after failures does not satisfy this geometric story. Repeated attempts can be dependent or have changing hazards, so an observed waiting count does not alone establish memorylessness or constant p.

The Poisson distribution models a nonnegative integer count with parameter λ≥0: p_X(k)=exp(−λ)λ^k/k!, k=0,1,…, with λ=0 interpreted as a point mass at0. Its normalisation uses the exponential-series identity Σλ^k/k!=exp(λ); the discrete route can use this stated identity without the calculus proof in Module16. A common generating story is a count in a fixed interval from a homogeneous process with independent increments and constant event rate, so λ is rate times interval length. This distribution is not automatically a model for every count, clustered process or varying-rate interval.

At λ=2, P(X=0)=exp(−2)≈.135335 and P(X=1)=2exp(−2)≈.270671. A truncated numerical table from0 through K has total less than one because it omits a positive tail, not because the model is invalid. Renormalising that truncated table changes the distribution to the conditional model X≤K. Lab 1 reports its missing mass explicitly. In very large-parameter computations direct powers and factorials can overflow; later stable log computations address arithmetic, while the support and generating assumptions remain unchanged.

Sampling without replacement from a fixed finite collection generally gives a hypergeometric count rather than a binomial one. The dependence from removing a token changes later success probabilities. A binomial approximation may be useful when the sampled fraction is small, but it must be identified as an approximation with an adequate error criterion. Named families are compact descriptions of mechanisms; matching a mean count or fitting a histogram shape is insufficient to establish their whole assumptions.

CS bridge: a PMF assigns probabilities directly to discrete values, while its CDF cumulates those masses with ≤. Practise endpoint identities and the categorical, binomial and geometric examples above. The rest of this section and all labelled continuous calculations require Module17; the CS route can omit them and continue with discrete quantiles and discrete transformations. A description of a continuous branch is not an extra hidden prerequisite for the discrete route.

Continuous AI branch: an absolutely continuous distribution has density f_X(x)≥0 with integral over R equal to one and probabilities P(a<X≤b)=∫_a^b f_X(x)dx. A PDF value is a height, not an event probability. It may exceed one, has units reciprocal to x, and its value at a single point can be changed without altering any probability. Continuous point probabilities are zero even at a positive density. Not every distribution has a PDF: a discrete atom or mixed model needs its point masses retained.

Uniform(a,b), a<b, has density1/(b−a) on(a,b), zero outside, and linear CDF inside. Whether density endpoint values are included does not alter its integrals. Exponential rate λ>0 has density λexp(−λx) for x≥0 and zero for x<0; its CDF on x≥0 is1−exp(−λx). Integrating the density gives one. Rate λ and scale1/λ are different conventions with different units: rate has inverse-time units, scale has time units. Memorylessness follows from the exponential tail product, with positive conditioning tail, and describes a constant-rate model rather than an inevitable property of all waiting times.

Gaussian N(μ,σ²), σ>0, has density exp(−(x−μ)²/(2σ²))/(σ√(2π)) on all R. Normalisation uses the standard Gaussian integral ∫exp(−z²/2)dz=√(2π), stated as a classical fact here rather than proved by the elementary antiderivative rules. Standardising z=(x−μ)/σ then supplies the scale factor. Its CDF generally has no elementary closed form and is computed numerically or via a special function. σ=0 is a degenerate atom at μ, not the formula obtained by dividing by zero.

Laplace(μ,b), b>0, has density exp(−|x−μ|/b)/(2b). Split its integral at μ: each exponential half contributes1/2. Its CDF is(1/2)exp((x−μ)/b) below μ and1−(1/2)exp(−(x−μ)/b) above. The density has a peak with a kink and heavier exponential tails than a Gaussian; selecting it as a noise model is an assumption about the error mechanism. Neither symmetric shape nor a few central observations uniquely determines which family applies.

A mixed distribution can place mass .3 at zero and distribute .7 uniformly on(0,1). Its CDF is0 below0, jumps to.3 at0, then is.3+.7t for0<t<1, and reaches1 at1. A density .7 on(0,1) represents only the continuous component and integrates to.7, so it cannot represent the entire probability model by itself. A dataset with a real point mass and continuous measurements requires a mixed representation; forcing either a pure PMF over all real observations or an ordinary PDF to include the atom loses information.

For absolutely continuous X, F is an integral of its density and F′=f where the appropriate regularity holds; a derivative at every point is unnecessary for representing the probabilities. For arbitrary distributions the CDF remains valid even when no ordinary density exists. This is why interval calculations through F are often safer than inferring a density from a smooth-looking plot. Keep the mathematical type of the distribution explicit before asking a numerical library for PMF or PDF values.

For0<u<1 define the generalised quantile Q(u)=inf{x:F(x)≥u}. The limit properties ensure its threshold is finite for these interior u values; right-continuity ensures the infimum has F(Q(u))≥u. The key equivalence Q(u)≤t exactly when u≤F(t) follows by monotonicity and right-continuity. A continuous strictly increasing CDF has an ordinary inverse, but a discrete step CDF needs this generalised definition. A quantile level is a probability, while its output is in the variable’s units.

The equivalence proves inverse-CDF sampling: if U is ideal uniform on(0,1), then P(Q(U)≤t)=P(U≤F(t))=F(t). On the discrete route, regard the construction as choosing a value by the lengths of its cumulative probability intervals; no density integration is required to implement the finite weighted sampler. Its correctness is the interval-mass identity. In a continuous branch the same argument uses the uniform interval model and yields the desired continuous distribution.

For a discrete table with sorted values x₁<…<x_m and cumulative masses c_j, choose the first j with c_j≥u and return x_j. The intervals(c_{j−1},c_j] have lengths p_j; zero masses contribute empty intervals and can be removed. With head-count masses1/4,1/2,1/4, Q(.2)=0, Q(.5)=1 and Q(.9)=2. Q(.25)=0 under this infimum convention. Quantile ties and endpoint conventions should be stated instead of relying on an implementation’s undocumented interpolation choice.

In the continuous branch, exponential inverse sampling solves u=1−exp(−λx), giving x=−log(1−u)/λ. Use log1p(−u) to compute log(1−u) accurately for small u, and exclude u=1 because the result is infinite. Affine uniform sampling uses a+(b−a)u. These formulas assume the input uniform model; a finite-precision pseudorandom generator actually samples from a finite representable grid. It approximates the ideal continuous mechanism, with repeatability and numerical details separate from the exact distribution theorem.

Given observations x₁,…,x_N, the empirical CDF is F_N(t)=(1/N)Σ_i 1{x_i≤t}. It is a discrete distribution assigning mass1/N per observation, with duplicate values collecting larger jumps. Counting sorted values using a right insertion index implements ≤; a left insertion index instead counts strictly smaller observations. An empirical CDF from continuous data is still stepwise, because a finite sample contains only finitely many observed values. Smoothing it produces another estimator, not the definition of the ECDF.

Compare empirical and theoretical CDFs at stated thresholds, rather than require finite samples to match exact masses. A seeded trace can illustrate sampling but not establish a universal discrepancy guarantee; Module24 adds relevant concentration and limit statements. Quantile interpolation in data software may differ from the distribution’s generalised inverse, especially with small samples and ties. Report the convention when a median, percentile or tail threshold feeds an algorithm or benchmark. The distribution definition and a numerical summary must agree on inclusions.

The explorer compares a binomial PMF with its step CDF, and, in the continuous branch, a uniform PDF with its CDF. It reports P(X=t), P(X≤t) and the appropriate mass or density separately. Switching modes changes the meaning of the plotted height. A PDF value of2 for uniform(0,.5) is compatible with total mass1 and zero point probability.

For Y=g(X), the event identity P(Y∈S)=P(X∈g⁻¹(S)) is the starting point. For discrete X, p_Y(y)=Σ_{x:g(x)=y}p_X(x). An injective transformation simply relabels masses, while a many-to-one transformation adds every preimage. Output support is the image of the original positive-mass support, not every number at which a formula can be evaluated. A transformed variable can have fewer possible values and remains dependent on the original variable.

Continuous AI branch: if g is a differentiable strictly monotone transformation with differentiable inverse on the relevant interval, then f_Y(y)=f_X(g⁻¹(y)) |(g⁻¹)′(y)| on the transformed support. Derive it from the CDF: for increasing g, F_Y(y)=F_X(g⁻¹(y)), so chain differentiation gives the inverse derivative. For decreasing g, the event becomes X≥g⁻¹(y); with no point mass, the complementary CDF gives the negative derivative. The absolute value unifies both cases and ensures nonnegative density.

For Y=cX+d with c≠0, f_Y(y)=f_X((y−d)/c)/|c|. The reciprocal factor balances the expansion or compression of interval lengths. X uniform(0,1) and Y=2X therefore has density1/2 on(0,2). Omitting1/2 yields total mass2. If c=0, Y is a point mass at d and this density formula does not apply; dividing by zero cannot represent a degenerate distribution. A Jacobian is about coordinate volume, not a change in total probability.

When g is not one-to-one, split its monotone branches and sum their inverse contributions, taking care with support and exceptional critical points. For X uniform(−1,1), Y=X² has F_Y(y)=P(−√y≤X≤√y)=√y for0≤y≤1. Differentiating on0<y<1 gives f_Y(y)=1/(2√y). The two branches ±√y each contribute1/(4√y); omitting one loses half the mass. The singular density near0 is integrable: ∫_0^1 1/(2√y)dy=1, and P(Y=0)=0.

This CDF-first derivation is often safer than memorising a single inverse formula. Outside the transformed support, probability and density are zero; at exceptional points one may need a separate atom or special treatment. A transformation that collapses a positive-probability interval to one value produces an atom, so not every transformation of an absolutely continuous variable stays absolutely continuous. For example max(X,0) for uniform(−1,1) has an atom of1/2 at0 and density1/2 on(0,1).

Check units as well as normalisation. If X measures seconds and Y=1000X milliseconds, the numeric density height shrinks by1000 because its units become inverse milliseconds; interval probabilities remain identical after unit conversion. Density values used as continuous likelihoods therefore depend on the observation unit. Likelihood ratios for two models transformed consistently share and cancel that coordinate factor, while comparing raw heights across inconsistent units is meaningless. Later estimation separates these density-based likelihoods from actual point probabilities.

Choose a family from a plausible data-generating story and check its support. A Bernoulli represents one binary event, a categorical variable one label, a binomial count fixed independent trials, a geometric count first success under stable repeated trials, and a Poisson count a specified rate mechanism. Continuous waiting times can use an exponential model under its memorylessness assumption; Gaussian or Laplace errors need an appropriate location, scale and tail story. Matching a plotted curve or knowing that a quantity is “random” does not supply these assumptions.

Support errors are often easier to detect than fitting errors. A trials-to-success variable cannot be0 under the stated convention; a failures-to-success variable can. An exact Poisson count is nonnegative and integer-valued, whereas a Gaussian permits negative and noninteger values. A Gaussian can sometimes be a useful count approximation in an appropriate regime, but then approximation error and boundary handling need separate justification. A uniform interval requires its endpoints in increasing order; an exponential rate and Gaussian standard deviation must be positive.

A probability implementation should state parameter units and names: rate versus scale, variance versus standard deviation, and geometric trials versus failures. Validate nonnegative categorical weights and a positive total before normalising; reject empty tables and invalid parameters. Verify finite distribution sums exactly when possible, and report tails for infinite prefixes. A small numerical normalisation error is different from a large missing branch, negative mass or wrong support. Explain the intended mathematical quantity before applying a numerical repair.

When probability is reported through a numerical CDF difference, cancellation can reduce accuracy for extremely short intervals or far tails. A dedicated survival-function calculation can represent a tail more accurately than subtracting a rounded near-one CDF from one. This arithmetic issue changes how a formula is evaluated, while the event identity remains the same. Record the threshold, inclusion convention and numerical method so a reported zero tail is not mistaken for a mathematically impossible event.

Labs use standard-library exact fractions for finite masses and local seeds for samples. Their outputs clearly label CS and continuous AI sections. CS learners inspect the discrete portions only and replace continuous practice with their preimage/CDF alternatives; AI learners use Module17 to derive and verify the displayed integrals. The simulations compare empirical CDFs at a few thresholds and make no unproved all-threshold accuracy promise. Generated Gaussian and exponential samples are numerical draws, not an exact continuum representation.

For learning objectives, discrete observation likelihoods use PMF values, while absolutely continuous observation likelihoods use PDF values under a chosen coordinate reference. A continuous density at a specific x is not the event probability P(X=x). Products over records also need their sampling factorisation assumptions. Module25 derives estimators and losses from these distinctions. Before then, your complete result is a derived distribution, a correctly included event probability, and a transformation whose total mass and support survive scrutiny.

Lab 1 · Discrete PMFs CDFs and infinite tails

Download lab1_discrete_distributions.py

"""Exact binomial/geometric tables; numerical Poisson partial sums."""
from fractions import Fraction as F
from math import comb, exp, factorial

if __name__ == "__main__":
    n, p = 5, F(2, 5)
    masses = [F(comb(n, k))*p**k*(1-p)**(n-k) for k in range(n+1)]
    cumulative = F(0)
    for k, mass in enumerate(masses):
        cumulative += mass
        print("binomial k=%d pmf=%s cdf=%s" % (k, mass, cumulative))
    assert cumulative == 1
    print("binomial P(X>=3)=%s = %.8f" % (sum(masses[3:]), float(sum(masses[3:]))))
    p = F(1, 4)
    K = 10
    partial = sum((p*(1-p)**(k-1) for k in range(1, K+1)), F(0))
    tail = (1-p)**K
    print("geometric trials support starts at1: sum1..10=%s remaining=%s" % (partial, tail))
    assert partial+tail == 1
    print("geometric P(X>3)=%s" % ((1-p)**3))
    rate = 2.0
    for K in (5, 10, 20):
        partial = sum(exp(-rate)*rate**k/factorial(k) for k in range(K+1))
        print("Poisson lambda=2 sum0..%d=%.12f remainder=%.3e" % (K, partial, 1-partial))
    print("PASS: finite normalisation exact; an infinite table needs an explicit tail")

binomial k=0 pmf=243/3125 cdf=243/3125
binomial k=1 pmf=162/625 cdf=1053/3125
binomial k=2 pmf=216/625 cdf=2133/3125
binomial k=3 pmf=144/625 cdf=2853/3125
binomial k=4 pmf=48/625 cdf=3093/3125
binomial k=5 pmf=32/3125 cdf=1
binomial P(X>=3)=992/3125 = 0.31744000
geometric trials support starts at1: sum1..10=989527/1048576 remaining=59049/1048576
geometric P(X>3)=27/64
Poisson lambda=2 sum0..5=0.983436391519 remainder=1.656e-02
Poisson lambda=2 sum0..10=0.999991691776 remainder=8.308e-06
Poisson lambda=2 sum0..20=1.000000000000 remainder=5.995e-15
PASS: finite normalisation exact; an infinite table needs an explicit tail

Predict the exact binomial tail, geometric omitted mass and Poisson prefix behaviour. Explain why finite binomial normalisation and an infinite Poisson prefix need different checks.

Lab 2 · Samples and empirical CDFs

Download lab2_sampling_and_ecdfs.py

"""Seeded discrete/continuous samples with exact CDF references; standard library."""
from bisect import bisect_right
from math import log1p, exp
import random


def geometric_trials(u, p):
    if not 0 < p <= 1 or not 0 <= u < 1:
        raise ValueError("require 0<p<=1 and0<=u<1")
    if p == 1:
        return 1
    # First k with 1-(1-p)^k >= u.
    import math
    return max(1, math.ceil(log1p(-u)/log1p(-p)))


if __name__ == "__main__":
    generator = random.Random(22022)
    N = 20000
    # The finite generator can return0: assign that endpoint to the first trial.
    geometric = sorted(geometric_trials(generator.random(), .25) for _ in range(N))
    print("CS branch: geometric trials, p=.25, N=20000")
    for t in (1, 3, 5, 10):
        empirical = bisect_right(geometric, t)/N
        exact = 1-.75**t
        print("t=%2d empirical CDF=%.6f exact=%.6f" % (t, empirical, exact))
    # AI branch uses Module17 integration to derive these analytic CDFs.
    exponential = sorted(-log1p(-generator.random())/2 for _ in range(N))
    normal = sorted(generator.gauss(0, 1) for _ in range(N))
    print("AI continuous branch: exponential rate2 and standard Gaussian")
    for t in (.25, .5, 1.):
        print("exponential t=%.2f empirical CDF=%.6f exact=%.6f" % (
            t, bisect_right(exponential, t)/N, 1-exp(-2*t)))
    from math import erf, sqrt
    for t in (-1., 0., 1.):
        print("Gaussian t=%+.1f empirical CDF=%.6f exact=%.6f" % (
            t, bisect_right(normal, t)/N, .5*(1+erf(t/sqrt(2)))))
    print("ECDF uses <= via bisect_right; a single trace is not an error theorem.")

CS branch: geometric trials, p=.25, N=20000
t= 1 empirical CDF=0.251550 exact=0.250000
t= 3 empirical CDF=0.583400 exact=0.578125
t= 5 empirical CDF=0.770100 exact=0.762695
t=10 empirical CDF=0.945750 exact=0.943686
AI continuous branch: exponential rate2 and standard Gaussian
exponential t=0.25 empirical CDF=0.393850 exact=0.393469
exponential t=0.50 empirical CDF=0.630350 exact=0.632121
exponential t=1.00 empirical CDF=0.863400 exact=0.864665
Gaussian t=-1.0 empirical CDF=0.161500 exact=0.158655
Gaussian t=+0.0 empirical CDF=0.499350 exact=0.500000
Gaussian t=+1.0 empirical CDF=0.841250 exact=0.841345
ECDF uses <= via bisect_right; a single trace is not an error theorem.

CS learners inspect the geometric section and its quantile; AI learners also derive exponential and Gaussian CDF references with Module17. Explain the ≤ convention, local seed and finite-trace limitation. Derive a discrete inverse sampler rather than spend CS study time on continuous formulas.

Lab 3 · Transformation and support faults

Download lab3_transformation_and_support_faults.py

"""Many-to-one discrete transforms, Jacobian correction, and support diagnostics."""
from fractions import Fraction as F
from math import sqrt, exp

if __name__ == "__main__":
    masses = {-2: F(1, 4), -1: F(1, 4), 1: F(1, 4), 2: F(1, 4)}
    transformed = {}
    for x, mass in masses.items():
        transformed[x*x] = transformed.get(x*x, F(0)) + mass
    print("CS branch X masses:", {x: str(p) for x, p in masses.items()})
    print("Y=X^2 masses (sum every preimage):", {y: str(p) for y, p in transformed.items()})
    assert sum(transformed.values()) == 1
    print("support diagnosis: waiting trials cannot be0; geometric failures can be0")
    print("Poisson counts include0; Gaussian counts can be negative/noninteger, so are not a count model")
    print("AI continuous branch requires Module17: X uniform(0,1), Y=2X")
    print("wrong density1 on(0,2): total mass=2; corrected density1/2: total mass=1")
    print("nonmonotone example X uniform(-1,1), Y=X^2")
    for y in (.04, .25, .81):
        cdf = sqrt(y)
        density = 1/(2*sqrt(y))
        print("y=%.2f CDF=%.6f density=%.6f" % (y, cdf, density))
    eps = 1e-4
    corrected_mass = 1-sqrt(eps)  # integral eps..1 of 1/(2sqrt(y))
    omitted_branch_mass = corrected_mass/2
    print("mass eps..1: both branches=%.6f one branch only=%.6f omitted0..eps=%.6f" % (
        corrected_mass, omitted_branch_mass, sqrt(eps)))
    assert abs(corrected_mass+sqrt(eps)-1) < 1e-12
    print("exponential rate2: density at0=2, P(X=0)=0, P(0<X<=.1)=%.8f" % (1-exp(-.2)))
    print("PASS: discrete preimages add; continuous inverse branches need absolute Jacobians")

CS branch X masses: {-2: '1/4', -1: '1/4', 1: '1/4', 2: '1/4'}
Y=X^2 masses (sum every preimage): {4: '1/2', 1: '1/2'}
support diagnosis: waiting trials cannot be0; geometric failures can be0
Poisson counts include0; Gaussian counts can be negative/noninteger, so are not a count model
AI continuous branch requires Module17: X uniform(0,1), Y=2X
wrong density1 on(0,2): total mass=2; corrected density1/2: total mass=1
nonmonotone example X uniform(-1,1), Y=X^2
y=0.04 CDF=0.200000 density=2.500000
y=0.25 CDF=0.500000 density=1.000000
y=0.81 CDF=0.900000 density=0.555556
mass eps..1: both branches=0.990000 one branch only=0.495000 omitted0..eps=0.010000
exponential rate2: density at0=2, P(X=0)=0, P(0<X<=.1)=0.18126925
PASS: discrete preimages add; continuous inverse branches need absolute Jacobians

The discrete branch sums square-map preimages and checks waiting/count supports. The continuous branch, requiring Module17, repairs the missing affine factor and omitted inverse branch, then distinguishes density2 from probability. Every output is captured from the displayed downloadable code.

Exercises 1–12 are required; branch alternatives occupy the same time. Optional Exercises 13–14 add35 minutes beyond the ten-hour core schedule.

Use random-variable and distribution materials in MIT 6.041SC and probability selections in the authors’ Mathematics for Machine Learning companion. The continuous branch uses Module17; CS readers select discrete PMFs, CDFs and transforms.

Distinguish outcome, map and realisation; derive one PMF; compute events with CDF endpoint conventions; specify a quantile sampler and a many-to-one transformation. AI learners additionally check density integrals and absolute inverse factors.

Exit task: derive the binomial tail992/3125, geometric trial quantile at .5 equal3, and a square-map distribution with both signs included. Explain which mathematical type each reported height represents.

Ready to move on: you can choose and transform distributions with stated assumptions. Expectation and joint models next connect these probabilities to average costs, variation and dependence. See the course overview.