This article possibly contains original research. (November 2023)
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 of random variables has a bivariate normal distribution means that every linear combination of and for constant (i.e. not random) coefficients and (not both equal to zero) has a univariate normal distribution. In that case, if and are uncorrelated then they are independent. However, it is possible for two random variables and 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.
A symmetric example
To see that and are uncorrelated, one may consider the covariance : by definition, it is
Then by definition of the random variables , , and , and the independence of from , one has
To see that has the same normal distribution as , consider
(since and both have the same normal distribution), where is the cumulative distribution function of the Standard normal distribution.
To see that and are not independent, observe that or that .
Finally, the distribution of the simple linear combination concentrates positive probability at 0: . Therefore, the random variable is not normally distributed, and so also and are not jointly normally distributed (by the definition above).
An asymmetric example
Suppose has a normal distribution with expected value 0 and variance 1. Let
where is a positive number to be specified below. If is very small, then the correlation is near if is very large, then is near 1. Since the correlation is a continuous function of , the intermediate value theorem implies there is some particular value of that makes the correlation 0. That value is approximately 1.54.[note 1] In that case, and are uncorrelated, but they are clearly not independent, since completely determines .
To see that is normally distributed—indeed, that its distribution is the same as that of —one may compute its cumulative distribution function:
where the next-to-last equality follows from the symmetry of the distribution of and the symmetry of the condition that .
In this example, the difference is nowhere near being normally distributed, since it has a substantial probability (about 0.88) of it being equal to 0. By contrast, the normal distribution, being a continuous distribution, has no discrete part—that is, it does not concentrate more than zero probability at any single point. Consequently and are not jointly normally distributed, even though they are separately normally distributed.
Examples with support almost everywhere in ℝ2
It is well-known that the ratio of two independent standard normal random deviates and has a Cauchy distribution. One can equally well start with the Cauchy random variable and derive the conditional distribution of to satisfy the requirement that with and independent and standard normal. Cranking through the math one finds that
in which is a Rademacher random variable and is a Chi-squared random variable with two degrees of freedom.
Consider two sets of , . Note that is not indexed by – that is, the same Cauchy random variable is used in the definition of both and . This sharing of results in dependences across indices: neither nor is independent of . Nevertheless all of the and are uncorrelated as the bivariate distributions all have reflection symmetry across the axes.
The figure shows scatterplots of samples drawn from the above distribution. This furnishes two examples of bivariate distributions that are uncorrelated and have normal marginal distributions but are not independent. The left panel shows the joint distribution of and ; the distribution has support everywhere but at the origin. The right panel shows the joint distribution of and ; the distribution has support everywhere except along the axes and has a discontinuity at the origin: the density diverges when the origin is approached along any straight path except along the axes.
- Hogg, Robert; Tanis, Elliot (2001). "Chapter 5.4 The Bivariate Normal Distribution". Probability and Statistical Inference (6th ed.). pp. 258–259. ISBN 0130272949.
- UIUC, Lecture 21. The Multivariate Normal Distribution, 21.6:"Individually Gaussian Versus Jointly Gaussian".
- Rosenthal, Jeffrey S. (2005). "A Rant About Uncorrelated Normal Random Variables".
- Edward L. Melnick and Aaron Tenenbein, "Misspecifications of the Normal Distribution", The American Statistician, volume 36, number 4 November 1982, pages 372–373
- More precisely 1.53817..., the square root of the median of a chi-squared distribution with 3 degrees of freedom.