Marginaalitiheys

Marginaalitiheys refers to the probability density function of a subset of random variables from a larger set of random variables. In simpler terms, it's the probability distribution of one or more variables when you ignore the others.

Imagine you have a system with multiple variables, say X, Y, and Z. If you are only interested in the probability distribution of X, you would calculate its marginal density by integrating the joint probability density function of X, Y, and Z over all possible values of Y and Z. This process effectively "averages out" the influence of Y and Z, leaving you with the distribution of X alone.

The concept of marginal density is fundamental in probability theory and statistics. It allows us to analyze the behavior of individual variables or subsets of variables within a complex system without needing to consider the entire system's joint distribution. This simplifies analysis and can reveal important characteristics of the individual components.

For continuous random variables, the marginal density is obtained by integration. For discrete random variables, it is obtained by summation. The marginal density function of a variable X, denoted as f_X(x), is derived from the joint probability density function f_{X,Y}(x,y) by integrating over all possible values of Y: f_X(x) = ∫ f_{X,Y}(x,y) dy. Similarly, for discrete variables, f_X(x) = Σ f_{X,Y}(x,y).