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In probability theory and related fields, a stochastic or random process is a random function, i. e. a mathematical object usually defined as a family of random variables. Historically, the random variables were associated with or indexed by a set of numbers, usually viewed as points in time, giving the interpretation of a stochastic process representing numerical values of some system randomly changing over time, such as the growth of a bacterial population, an electrical current fluctuating due to thermal noise, or the movement of a gas molecule.

The study of stochastic processes uses mathematical knowledge and techniques from probability, calculus, linear algebra, set theory, and topology as well as branches of mathematical analysis such as real analysis, measure theory, Fourier analysis, and functional analysis. The theory of stochastic processes is considered to be an important contribution to mathematics and it continues to be an active topic of research for both theoretical reasons and applications.

## Examples

One of the simplest stochastic processes is the Bernoulli process, which is a sequence of independent and identically distributed (i. i. d.) random variables, where each random variable takes either the value one or zero, say one with probability $p$ and zero with probability $1-p$. This process can be linked to repeatedly flipping a coin, where the probability of obtaining a head is $p$ and its value is one, while the value of a tail is zero. In other words, a Bernoulli process is a sequence of i. i. d. Bernoulli random variables, where each coin flip is an example of a Bernoulli trial.

As random gameplay elements in Asphalt are usually about getting a desired item or not (1 or 0), all of them are stochastic processes, and, from this point of view, most of them are also Bernoulli processes:

• Opening random containers like Pro Kit Boxes (only a Bernoulli process if all contained cards have the same possible outcomes (sample space) with the same probabilities. This is not the case if there are guarantees for a specific card, for instance "at least one card will be rare": This card can never be common, so it has a different probability distribution and is therefore a separate process.)
• getting rewards for daily ads
• getting rewards for Daily Tasks
• Multiplayer matchmaking (Bernoulli process if a specific opponent is wanted),
• Multiplayer track selection (Bernoulli process if a specific track is wanted),
• the distribution of Quality checks and Festival coins on race tracks (can be a Bernoulli process when it is observed if an item appears at a specific place or not)
• the color of AI opponents in races (only a Bernoulli process if the occurence of a specific color of a specific vehicle is observed)

One of the rare examples in the game where the result of an experiment can have more than two countable outcomes are the Bundles of the 5th Anniversary festival. The Expert Bundle, for example, grants between 1 and 7 Blueprints for the Audi R8 e-tron Special Edition. As such, observing the number of granted blueprints is a separate stochastic process within the box that has 7 possible outcomes. However, if the occurence of a specific number of blueprints (for example, 4) is observed, opening Expert Bundles becomes a Bernoulli process with the two outcomes of getting 4 blueprints or not.

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