Courses / Mathematics / Calculus-based probability: laws, dependence and limits Mathematics · unit version 1 Calculus-based probability: laws, dependence and limits Lessons develop discrete and continuous laws, transformations, multivariate probability, expectation and limiting arguments. Original exercises include numerical checks and written proofs. This sequence is not a claim of complete independently assessed probability-course coverage; measure-theoretic foundations, stochastic processes and advanced convergence remain further work.
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Readiness and preparation Check your starting point Try these short questions on paper, then compare your reasoning. They are optional self-checks, not a placement score or a navigation requirement. Use a targeted review when useful and return with your browser’s Back control.
If P(A)=0.3 and P(A∩B)=0.12, what is P(B|A)? Does this determine P(A|B)? Conditioning changes the reference population and requires a positive denominator.
Compare reasoning and find a refresher P(B|A)=0.12/0.3=0.4. Reversing the conditional requires P(B), which is not supplied; joint probability alone is not enough.
Review conditional probability
Normalize a density f(x)=cx on 0<x<2. Why is f(1) not P(X=1)? Continuous laws assign probability through integrals, not density heights.
Compare reasoning and find a refresher The integral is c·[x²/2]₀²=2c, so c=1/2. A single point has zero integral and hence probability zero; f(1)=1/2 is a density per unit.
Review definite integration
How do “for every outcome there exists a bound” and “there exists a bound for every outcome” differ? Probability limits and uniform bounds depend on the order of quantifiers.
Compare reasoning and find a refresher In the first statement the bound may depend on the outcome. The second asks for one bound that works simultaneously for all outcomes, which is stronger.
Review quantifier dependence
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Probability: laws, dependence and justified limits
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Probability investigation: arrival dependence, route termination and incomplete evidence
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Lesson sequence Each linked lesson opens on its own page. Expand its details here when you want to check the objectives before choosing.
Lesson details and objectives Use countable additivity rather than intuition about infinite lists.
Apply complements, unions and continuity of probability. Distinguish countable from uncountable unions. Relate increasing and decreasing event sequences to limits. 6 practice tasks.
Lesson details and objectives Separate conditioning, disjointness and independence of collections.
Use conditional multiplication with positive conditioning probability. Check mutual rather than only pairwise independence. Explain why conditioning can change dependence. 8 practice tasks.
Lesson details and objectives Compute joint branches and normalize only after adding all sources.
Apply total probability over a disjoint exhaustive partition. Derive a posterior from joint probabilities. Audit base-rate and zero-evidence conditions. 6 practice tasks.
Lesson details and objectives Connect events, point masses and interval probabilities.
Construct a CDF from a discrete law. Use left limits to include atoms correctly. State monotonicity, right continuity and endpoint limits. 6 practice tasks.
Lesson details and objectives Specify what is counted and the stopping convention.
Match a trial mechanism to a discrete family. Derive geometric tails and memorylessness. Distinguish count of failures from trial index of first success. 6 practice tasks.
Lesson details and objectives Derive the stopped-trial law, its moments and the exact effect of a recording cap.
Derive total-trial and failure-count mass functions with explicit supports. Prove normalization and moments through independent geometric waiting gaps. Use the count/stopping duality while retaining censored outcomes. 6 practice tasks.
Lesson details and objectives Connect ordered records, count vectors, covariance and changed collection protocols.
Derive multinomial masses and normalization from sequence counts. Prove the singular count covariance and handle structural-zero categories. Derive merging/conditional laws and compare copied-record protocols. 6 practice tasks.
Lesson details and objectives Separate a count distribution from stronger process assumptions.
Normalize the Poisson mass and compute probability ratios. State independent-increment assumptions when adding interval counts. Explain the binomial rare-event limit. 6 practice tasks.
Lesson details and objectives Treat density as a rate per unit, not a point probability.
Normalize nonnegative densities on their support. Compute CDFs and event probabilities by integration. Distinguish density height from mass. 6 practice tasks.
Lesson details and objectives Connect interval length, waiting-time survival and Gaussian standardization.
Use uniform length ratios and exponential survival. Explain memorylessness and rate units. Standardize a normal law using supplied tail values. 6 practice tasks.
Lesson details and objectives Use sums, integrals and integrability conditions.
Apply LOTUS to functions of a random variable. Distinguish E(g(X)) from g(E(X)). Recognize when an expectation fails to exist. 6 practice tasks.
Lesson details and objectives Connect squared deviations, moments and tail integrals.
Derive variance from first and second moments. Use survival integrals for nonnegative variables. Track moments under affine transformations. 6 practice tasks.
Lesson details and objectives Use CDF arguments and absolute derivative factors.
Derive monotone transformed densities. Sum contributions from nonmonotone inverse branches. Track support, atoms and singular endpoints. 6 practice tasks.
Lesson details and objectives Integrate out coordinates without replacing a triangle by a rectangle.
Compute marginal masses and densities. Translate joint events into integration regions. Check normalization on coupled supports. 6 practice tasks.
Lesson details and objectives Normalize each slice and avoid a zero-probability event ratio.
Derive conditional masses or densities on their support. Explain conditioning on continuous coordinates via densities. Compute conditional probabilities and expectations. 6 practice tasks.
Lesson details and objectives Test independence and construct uncorrelated dependent examples.
Check density or mass factorization on the full support. Compute covariance and correlation when moments exist. Distinguish independence from zero covariance. 6 practice tasks.
Lesson details and objectives Separate linearity of means from additivity of variances.
Derive variance of a linear combination. Use independence or covariance assumptions explicitly. Interpret diversification and perfect dependence. 8 practice tasks.
Lesson details and objectives Use the tower property with a well-defined conditioning law.
Construct E(X|Y) as a function of Y. Apply the tower property under integrability. Distinguish a conditional mean from an individual outcome. 6 practice tasks.
Lesson details and objectives Connect total variance with the best squared-error predictor.
Apply the law of total variance. Distinguish average conditional variance from variance of conditional means. Prove conditional expectation minimizes mean squared error. 6 practice tasks.
Lesson details and objectives Derive supports and Jacobians before evaluating integrals.
Construct discrete or continuous convolutions. Determine sum supports and piecewise integration limits. Use a multivariable inverse map with its absolute Jacobian. 6 practice tasks.
Lesson details and objectives Differentiate or identify an MGF with explicit existence conditions.
Derive basic MGFs and factor independent sums. State a neighborhood-of-zero uniqueness condition. Distinguish finite moments from existence of an MGF. 6 practice tasks.
Lesson details and objectives Use Markov, Chebyshev and Jensen before applying iid limit theorems.
Apply moment inequalities with their hypotheses. Derive a finite-variance weak law and distinguish it from the CLT. Check iid, variance and approximation conditions rather than assume a universal sample threshold. 8 practice tasks.
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