Randommultiples

Randommultiples is a term used to describe a random variable that yields values which are integer multiples of a fixed base number. Formally, let b be a nonzero integer. If M is a discrete random variable taking integer values, then X = b · M is a randommultiple with support determined by the support of M. For example, if M is uniformly distributed over {m1, m2, ..., mk}, then X is uniformly distributed over {b·m1, b·m2, ..., b·mk}.

Generation and variants: To generate a randommultiple, one selects a multiplier M according to a specified distribution and multiplies by the base b. Common choices include uniform distributions over a finite interval, geometric or Poisson distributions, or cryptographically secure RNGs. Variants may restrict to positive multiples, allow negative values, or confine values to a particular congruence class modulo another number.

Properties: If X = b · M, then the mean and variance satisfy E[X] = b · E[M] and Var(X) = b^2 · Var(M). When drawing independent samples X1, X2, ..., independence follows from the independence of the underlying multipliers M1, M2, ....

Applications: Randommultiples appear in simulations and randomized algorithms where numbers with a divisibility structure are desirable. They are used for testing arithmetic routines, sampling across multiples of a base period, or in procedural content generation where sizes or times are constrained to multiples of a unit.

Examples: With base b = 7 and M uniform on {0, 1, 2, 3}, X takes values 0, 7, 14, 21. With M uniform on {−2, −1, 0, 1, 2}, X takes −14, −7, 0, 7, 14.