Mathematical notation

This page defines the notation used across the dataset-creation documentation. Equations use descriptive \text{parameter name} labels instead of single Greek letters so readers can connect a formula directly to configuration names.

Probability

$P(A)$

The probability that event $A$ occurs. If $A$ means “a visitor buys,” then $P(A)=0.03$ means a 3 percent chance.

$P(A\mid B)$

The probability that $A$ occurs given that condition $B$ is already known. For example, $P(\text{purchase}\mid \text{click})$ is the purchase probability among people who clicked.

$A,B$

A comma between events means “and.” Thus $P(A,B)$ is the probability that both events occur.

Functions

$\exp (x)$

Raise the mathematical constant $e$ to the power $x$. The simulator uses this function to turn unrestricted normal noise into a positive multiplier.

$\ln (x)$

The natural logarithm. It reverses $\exp (x)$ and converts a positive multiplier into an additive log-odds adjustment.

$\mathrm{logit}(p)$

Convert a probability $p$ into log odds:

$$\mathrm{logit}(p)=\ln \left(\frac{p}{1-p}\right)$$

$\mathrm{logistic}(x)$

Convert log odds back into a probability:

$$\mathrm{logistic}(x)=\frac{1}{1+\exp (-x)}$$

This guarantees an answer between zero and one.

Sampling and aggregation

$\sim$

“Is sampled from” or “follows a distribution.” It describes randomness, not exact equality.

$\mathrm{Normal}(\text{mean},\text{standard deviation})$

A bell-shaped distribution centered on its mean. The standard deviation controls how spread out the draws are.

$\mathrm{Binomial}(\text{trials},\text{success probability})$

A count of successes from a fixed number of yes-or-no trials. For example, purchases can be sampled from 100 clicks with a 3 percent success probability.

$\sum$

Add all listed contributions. A label below the symbol tells the reader which items are included.

$\prod$

Multiply all listed factors.

$|\text{set}|$

The number of items in a set.

$!$

Factorial. For example, $3!=3\times 2\times 1=6$. It appears when counting possible touchpoint orderings for exact Shapley calculations.

Rounding and bounds

$max(a,b)$ and $min(a,b)$

Choose the larger or smaller value. The generator uses these functions to keep counts and standard deviations non-negative.

$\mathrm{round}(x)$

Round a continuous result to an integer count or a fixed number of currency decimals.

Reading descriptive equations

Text inside an equation names a configuration field or an intermediate quantity. For example:

$$\text{local unit price}=\text{base product price}\times \text{price multiplier}$$

This is the same calculation a program would express with base_product_price * price_multiplier; the mathematical version is for explanation, while the code name remains exact.