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When the image (or range) of is finitely or infinitely countable, the random variable is called a discrete random variable [5]: 399 and its distribution is a discrete probability distribution, i.e. can be described by a probability mass function that assigns a probability to each value in the image of .
where is the Dirac measure and are random variables, is called a point process [1] [2] or random counting measure. This random measure describes the set of N particles, whose locations are given by the (generally vector valued) random variables X n {\displaystyle X_{n}} .
A random process is a sequence of random variables whose outcomes do not follow a deterministic pattern, but follow an evolution described by probability distributions. These and other constructs are extremely useful in probability theory and the various applications of randomness .
These correspond to aggregates of random variables described using graphical models, where individual random variables are linked in a graph structure with conditional distributions relating variables to nearby variables. Multilevel models are subclasses of Bayes networks that can be thought of as having multiple levels of linear regression.
Random variables are usually written in upper case Roman letters, such as or and so on. Random variables, in this context, usually refer to something in words, such as "the height of a subject" for a continuous variable, or "the number of cars in the school car park" for a discrete variable, or "the colour of the next bicycle" for a categorical variable.
Another useful concept is mutual information defined on two random variables, which describes the measure of information in common between those variables, which can be used to describe their correlation. The former quantity is a property of the probability distribution of a random variable and gives a limit on the rate at which data generated ...
All three measures have the following property: If the random variable (or each value from the sample) is subjected to the linear or affine transformation, which replaces X by aX + b, so are the mean, median and mode. Except for extremely small samples, the mode is insensitive to "outliers" (such as occasional, rare, false experimental readings ...
For instance, a risk-neutral measure is a probability measure which assumes that the current value of assets is the expected value of the future payoff taken with respect to that same risk neutral measure (i.e. calculated using the corresponding risk neutral density function), and discounted at the risk-free rate.