🔗 Normally distributed and uncorrelated does not imply independent

🔗 Statistics

In probability theory, although simple examples illustrate that linear uncorrelatedness of two random variables does not in general imply their independence, it is sometimes mistakenly thought that it does imply that when the two random variables are normally distributed. This article demonstrates that assumption of normal distributions does not have that consequence, although the multivariate normal distribution, including the bivariate normal distribution, does.

To say that the pair ${\displaystyle (X,Y)}$ of random variables has a bivariate normal distribution means that every linear combination ${\displaystyle aX+bY}$ of ${\displaystyle X}$ and ${\displaystyle Y}$ for constant (i.e. not random) coefficients ${\displaystyle a}$ and ${\displaystyle b}$ has a univariate normal distribution. In that case, if ${\displaystyle X}$ and ${\displaystyle Y}$ are uncorrelated then they are independent. However, it is possible for two random variables ${\displaystyle X}$ and ${\displaystyle Y}$ to be so distributed jointly that each one alone is marginally normally distributed, and they are uncorrelated, but they are not independent; examples are given below.