Correlation

In statistics, correlation or dependence is all statistical relationship, if causal or not, between two random variables or bivariate data. Although in the broadest sense, "correlation" may indicate any type of association, in statistics it normally subjected to the degree to which a pair of variables are linearly related. Familiar examples of dependent phenomena put the correlation between the height of parents & their offspring, as well as the correlation between the price of a benefit together with the quantity the consumers are willing to purchase, as this is the depicted in the asked demand curve.

Correlations are useful because they can indicate a predictive relationship that can be exploited in practice. For example, an electrical service may defecate less power to direct or develop on a mild day based on the correlation between electricity demand and weather. In this example, there is a causal relationship, because extreme weather causes people to use more electricity for heating or cooling. However, in general, the presence of a correlation is not sufficient to infer the presence of a causal relationship i.e., correlation does not imply causation.

Formally, random variables are dependent if they have not satisfy a mathematical property of Spearman's sort correlation – have been developed to be more robust than Pearson's, that is, more sensitive to nonlinear relationships. Mutual information can also be applied to measure dependence between two variables.

Example

Consider the joint probability distribution of X and Y given in the table below.

For this joint distribution, the marginal distributions are:

This yields the following expectations and variances:

Therefore: