For background on probability systems, see my previous post. A random variable is really a function, called a **measure function**, from the **sample space** to the **state space** of a probability system. Like , a state space is a -algebra, typically the space of real numbers . In this case a random variable is defined as , for a probability space . The notation will be used as shorthand for the more explicit, well defined .

Any real-valued random variable (e.g. ) can be fully characterized by its **cumulative distribution function**. The cumulative distribution function, or **CDF**, is defined for a random variable as . If the CDF is a continuous function, then the random variable is said to be a **continuous random variable**. This type of CDF is illustrated in Figure 1. For a continuous random variable the probability of taking on any particular value is infinitesimally small (i.e. ), so we talk about the probability of the random variable taking on a value in a particular range instead, . The probability of a random variable taking on a value in a particular range is called the probability density, and is represented by the **probability density function**, , illustrated in Figure 2. Note that can be estimated by using a sufficiently small interval,

For **discrete random variables**, the variable takes on particular values over a discrete interval. We can define a function that represents the probability that is exactly equal so to some value , or . The CDF is a step function since increases in discrete steps as shown in Figure 3. The probability that takes on a particular value is given by the **probability mass function**, illustrated in Figure 4. Notice that the pmf is not continuous, .

## 3 thoughts on “Random Variables and Distributions”

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