Learning objectives
By the end you can:
- State and apply the basic rules of probability (the addition rule (mutually exclusive and general), the multiplication rule, and conditional probability) to a two-event finance scenario, and correctly distinguish "mutually exclusive" from "independent."
- Compute the expected value, variance, and standard deviation of a discrete random variable, and apply the algebra of expectation and variance to a linear combination of two random variables, including the covariance cross-term that a two-asset portfolio's variance is built from.
- Derive Bayes' theorem from the definition of conditional probability, and apply it to invert a screen's sensitivity, false-positive rate, and base rate into a posterior probability, explaining, from the arithmetic itself, why a low base rate can make even a good screen's positive result far weaker evidence than it feels.
- Name and use six distributions (binomial, normal, lognormal, t, chi-square, F) stating what each models, its mean and variance, its shape, and the finance scenario each is the natural tool for.
- Explain the relationship between the normal distribution (of a continuously compounded return) and the lognormal distribution (of the resulting price level), and compute a probability or a percentile under each.
- State the Central Limit Theorem precisely, explain why it holds regardless of the shape of the underlying population (given a large-enough sample), and use it to describe the sampling distribution of a sample mean and compute its standard error.
- Distinguish a point estimator from an interval estimate, describe at an intuitive level what makes an estimator "good" (unbiasedness, efficiency, consistency), and explain why the sample-variance formula divides by n − 1.
- Construct a confidence interval for a mean (both z-based and t-based), for a proportion, and for a variance, and state, correctly, what a confidence level does and does not mean.
- (Productivity objective: R10 duality.) Dissect a confident, AI-produced statistical claim, a critical value, a Bayes' theorem set-up, end to end on both an Indian (₹) and a US ($) example, applying the Primary-Source Guardrail, and explain why quantitative methods carries this program's highest AI-over-trust risk.
- State precisely what this module hands off, and to where, hypothesis testing, p-values, and regression to QM1.02; every "in code" build to DA1.01/DA1.05 in the quant-code region, and explain, in one sentence each, why neither hand-off is repeated here.
The duality, stated once (R10). As in every module of this program, two objectives run in parallel. The understanding objective (items 1–8 and 10) is what the mastery gate rewards; you earn it by deriving the mechanism and running the by-hand computation yourself. The productivity objective (item 9) is the discipline of using AI to move faster on the arithmetic and the lookup while never letting it own the judgment call of which distribution applies or what a result means. Nowhere in this program is that distinction sharper, or more dangerous to blur, than here.
Prerequisites & connections
Builds on. Almost nothing, formally. Like this Ring's other opening nodes, QM1.01 is the first module of its branch and unlocks from day one with no finance-phase prerequisite. You need ordinary algebra (comfortable rearranging an equation, comfortable with exponents and natural logarithms) and enough arithmetic ease that a decimal doesn't intimidate you and a percentage isn't exotic. If you have already built the ratio discipline of Phase 2 (M2.01–M2.05) (define the number, compute it, sanity-check it) that habit will feel at home here, but nothing assumes you have done it. EX2.03's SLOPE, CORREL, and LINEST give you an Excel-native first look at correlation and regression output; useful context for QM1.02 later, not required for this node.
Feeds forward. Everything downstream of this Ring's quantitative spine leans on this module without re-deriving it (R1): carry the map: