AI and ML Mathematics

Calculate ratios, rates, proportions, and weighted averages from AI system measurements.

Rearrange a formula to isolate a desired system quantity.

Represent resource limits and performance requirements with inequalities.

Model fixed and variable costs with a linear function, interpreting its slope and intercept.

Apply exponent laws to simplify expressions involving repeated growth.

Calculate how a branching process grows exponentially with its depth.

Use logarithms to solve for an unknown exponent or recursion depth.

Compare linear, polynomial, and exponential growth using tables and plots.

Identify the domain, range, inputs, and outputs of a function used in a system model.

Compose functions to represent a multi-stage AI pipeline.

Define and evaluate a piecewise function for a routing or fallback rule.

Treat a symbol as a tunable parameter and predict how changing it alters a function's graph and modeled system behavior.

Represent an ordered sequence with indexed notation.

Generate terms of a recurrence relation from its initial condition and update rule.

Derive the finite geometric-series formula for total calls in a regular recursive workflow.

Plot a function or recurrence with provided Python code and connect the graph to system behavior.

Compute an average rate of change from a formula or table of measurements.

Interpret a derivative as local sensitivity using a graph and a system example.

Differentiate sums of power functions using the power and constant-multiple rules.

Differentiate basic exponential and logarithmic functions.

Apply the chain rule to a two-stage composition of functions.

Compute a partial derivative of a function with several inputs.

Assemble partial derivatives into a gradient and interpret its components.

Approximate a derivative with finite differences in Python and compare it with an analytic derivative.

Represent features, embeddings, or system state as a vector with labeled dimensions.

Add and scale vectors and interpret the resulting transformation.

Calculate vector norms and distances between vectors.

Calculate a dot product and interpret it as directional alignment.

Calculate cosine similarity and distinguish it from Euclidean distance.

Construct a weighted sum of vectors and interpret the role of the weights.

Express one vector as a linear combination of other vectors in a simple example.

Use NumPy arrays to reproduce vector calculations and visualize two-dimensional vectors.

Determine the shape and meaning of a matrix from its rows, columns, and indices.

Compute a matrix-vector product and trace how each output is formed.

Compute a small matrix-matrix product and verify that its dimensions are compatible.

Interpret a matrix as a linear transformation of vectors.

Use a transpose to reorganize dimensions and express batches of dot products.

Identify the axes and shapes of tensors representing batches, tokens, and features.

Interpret matrix rank as the number of independent directions represented by a transformation.

Trace tensor shapes through a simplified neural-network or attention computation.

Define a sample space and calculate probabilities of simple workflow events.

Represent possible values and probabilities with a discrete random variable.

Calculate conditional probability from a table or event diagram.

Apply Bayes’ rule to update a probability after observing evidence.

Determine whether an independence assumption is justified in a stated system scenario.

Calculate the expected value, variance, and standard deviation of a small discrete distribution and interpret them as measures of center and uncertainty.

Interpret the mean, standard deviation, probability density, and area under a normal curve, and standardize a value with a z-score.

Simulate repeated random outcomes in Python and compare empirical frequencies with theoretical probabilities.

Summarize repeated AI-system measurements using the mean, median, quantiles, and variability.

Distinguish a sample statistic from the population quantity it estimates.

Construct and interpret a bootstrap confidence interval using guided resampling or provided Python code.

Given a null hypothesis and a simulated or reference null distribution, calculate and interpret a p-value while distinguishing statistical significance from practical importance.

Select an evaluation metric for a stated task and identify a behavior that the metric could conceal.

Distinguish correlation from causation and identify a plausible confounding variable.

Design a controlled comparison of two prompts, models, or agent configurations.

Estimate the reliability and expected cost of a stochastic workflow from repeated trials.

Translate a training or system-design goal into decision variables, a scalar objective or loss, and constraints.

Calculate squared-error and absolute-error losses for a set of predictions.

Use a gradient to perform one step of gradient descent by hand.

Predict how an excessively small or large learning rate affects optimization.

Compare full-batch and stochastic gradient estimates on a small example.

Diagnose overfitting from training and validation results and explain the role of regularization.

Optimize a simple parameterized model in Python and visualize its loss over successive updates.

Calculate the trainable parameter counts for a full matrix update and a LoRA-style low-rank update.

Factor a sequence probability into conditional token probabilities using the probability chain rule.

Convert a small set of logits into probabilities with the softmax function.

Calculate the likelihood and negative log-likelihood of an observed token sequence.

Calculate entropy and cross-entropy for small probability distributions and explain their difference.

Calculate KL divergence for two small distributions and interpret its directionality.

Calculate perplexity from an average negative log-likelihood and interpret the result.

Predict how temperature, top-k, and top-p settings alter a token distribution.

Simulate token sampling in Python and relate observed variation to decoding settings.

Represent token embeddings as a sequence matrix and label its dimensions.

Trace the dimensions of query, key, and value projections in a single attention head.

Calculate scaled dot-product attention scores for a small worked example.

Convert attention scores into attention weights using softmax.

Compute an attention output as a weighted sum of value vectors.

Apply a causal mask to determine which token-to-token interactions are permitted.

Model prompting and in-context learning as conditioning on context, and contrast this with changing model weights.

Design a controlled experiment that tests one prompting technique while holding competing explanations constant.

Represent an agent workflow as a directed graph or state machine with deterministic and stochastic transitions.

Derive a worst-case bound on calls, tokens, or cost in a recursive agent workflow using recurrence and geometric-growth results.

Calculate expected calls, tokens, or cost when workflow branches have different probabilities.

Calculate end-to-end latency for workflows with sequential and parallel branches by identifying the critical path.

Derive end-to-end success probability under an explicit independence assumption and explain how dependencies could invalidate the result.

Identify feedback loops and formulate a measurable stopping condition for an iterative workflow.

Map a paper's variables, assumptions, claimed mechanism, and evidence onto a proposed system design, identifying one supported and one unsupported transfer claim.

Draft a concise experiment specification that connects a paper-derived idea to a system change, measurable outcomes, and decision criteria.