Thus it can also be seen as a family of random variables indexed by time. Example 6 In the experiment of ipping a coin once, the random variable given by X(H) = 1;X(T) = 1 represents the earning of a player who receives or loses an euro according as the outcome is heads or tails. A stochastic process is by nature continuous; by contrast a time series is a set of observations indexed by integersl. = ˆ 1 if !2A 4.

One of the main application of Machine Learning is modelling stochastic processes. Poisson processes:for dealing with waiting times and queues. Gaussian Processes:use… A stochastic process may involve several related random variables. Markov decision processes:commonly used in Computational Biology and Reinforcement Learning. Typical examples are the size of a population, the boundary between two phases in an alloy, or interacting molecules at positive temperature. Some examples of stochastic processes used in Machine Learning are: 1. No reason to only consider functions deﬁned on: what about functions ?

This random variable is discrete with P(X= 1) = P(X= 1) = 1 2: Example 7 If Ais an event in a probability space, the random variable 1 A(!) In probability theory and related fields, a stochastic or random process is a mathematical object usually defined as a family of random variables. 2. Example: Poisson process, rate . Stochastic Processes A sequence is just a function. A sequence of random variables is therefore a random function from . 3. Random Walk and Brownian motion processes:used in algorithmic trading. A stochastic process is a process evolving in time in a random way.