![[Pasted image 20240629141535.png|300]] # Random Variables **Randomness** can be described as the *presence of uncertainty* (or the absence of predictability). A random variable is a mathematical object takes in an element in a sample space and outputs a number The relationship between these two is described through a [[Probability Distributions 1|probability distribution]] A **random variable** will always have an associated probability distribution that represents the likelihood that any of the possible values would occur Random variables are known as *stochastic processes*. The term “stochastic” refers to the property of being well-described by a random [[Probability Distributions 1|probability distribution]]. - An *uppercase* random variable *has not been observed* yet and a *lowercase* random variable is associated with an observed value and is *no longer random*. > [!NOTE]+ Random Variables vs Events > > Random varaibles and events are two different concepts. > > An event is an outcome, or more often a set of outcomes, to an experiment. A random variable is more like an experiment --- Once a value has actually been determined for a random variable, we say that it has been **realized** (or observed). The value produced is known as a *realization* (or observation/observed value). ## Discrete vs Continuous Variables > See also: > - [[Probability Distributions 1]] ### Discrete Random Variables - [ ] Probability Mass Functions ### Continuous Random Variables In the case of **continuous random variables**, we can no longer talk about the probability that we observed $O$ given $\theta$, because at any given $O$, $P(O|\theta)=0$. This occurs because there are now an *infinite amount of possible outcomes* due to the nature of continuous functions. > [!summary] **Definition:** Continuous Random Variables > a > a > --- The distinction between continuous and discrete variables is blurred/nonexistant when viewed from the field of [[Measure Theory]]. ## Marginal Probabilities > See also: > - [[Joint Probability Distributions]] Given some subset of random variables from a larger collection, the marginal distribution The marginal distribution of a subset of a collection of random variables **Marginal Probability Mass Function** A marginal distribution gives the proba It contrasts with the