Continuous distribution in the context of Conditional probability distribution

In probability theory and statistics, the conditional probability distribution is a probability distribution that describes the probability of an outcome given the occurrence of a particular event. Given two jointly distributed random variables ${\displaystyle X}$ and ${\displaystyle Y}$, the conditional probability distribution of ${\displaystyle Y}$ given ${\displaystyle X}$ is the probability distribution of ${\displaystyle Y}$ when ${\displaystyle X}$ is known to be a particular value; in some cases the conditional probabilities may be expressed as functions containing the unspecified value ${\displaystyle x}$ of ${\displaystyle X}$ as a parameter. When both ${\displaystyle X}$ and ${\displaystyle Y}$ are categorical variables, a conditional probability table is typically used to represent the conditional probability. The conditional distribution contrasts with the marginal distribution of a random variable, which is its distribution without reference to the value of the other variable.

If the conditional distribution of ${\displaystyle Y}$ given ${\displaystyle X}$ is a continuous distribution, then its probability density function is known as the conditional density function. The properties of a conditional distribution, such as the moments, are often referred to by corresponding names such as the conditional mean and conditional variance.