TeXipedia

prod

Represents a mathematical product operation, used to multiply a sequence of terms or express repeated multiplication.

Overview

Essential in mathematical notation for expressing the multiplication of multiple terms, particularly in sequences, series, and indexed expressions.

  • Commonly used in combinatorics and number theory for factorial-like calculations
  • Appears frequently in probability theory for joint probabilities
  • Useful in expressing solutions to recurrence relations and infinite products
  • Often paired with subscripts and superscripts to define the range of multiplication

Examples

Computing the product of sequence elements from 1 to n

∏i=1ni=n!\prod_{i=1}^{n} i = n! 
\prod_{i=1}^{n} i = n!

Expressing the product of matrix elements

∏i=1naii\prod_{i=1}^{n} a_{ii} 
\prod_{i=1}^{n} a_{ii}

Calculating probability of independent events

P(A∩B∩C)=∏i=13P(Ei)=P(A)⋅P(B)⋅P(C)P(A \cap B \cap C) = \prod_{i=1}^{3} P(E_i) = P(A) \cdot P(B) \cdot P(C) 
P(A \cap B \cap C) = \prod_{i=1}^{3} P(E_i) = P(A) \cdot P(B) \cdot P(C)