Students of statistics and probability theory sometimes develop misconceptions about the normal distribution, ideas that may seem plausible but are mathematically untrue. For example, it is sometimes mistakenly thought that two linearly uncorrelated, normally distributed random variables must be statistically independent. However, this is untrue, as can be demonstrated by counterexample. Likewise, it is sometimes mistakenly thought that a linear combination of normally distributed random variables will itself be normally distributed, but again, counterexamples prove this wrong.

of random variables has a bivariate normal distribution means that every linear combination (not both equal to zero) has a univariate are uncorrelated then they are independent. 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.

Examples

A symmetric example

has a normal distribution with expected value have the Rademacher distribution are uncorrelated, as can be verified by calculating their covariance are not independent.

.

are not jointly normally distributed (by the definition above).

An asymmetric example

has a normal distribution with expected value is very small, then the correlation is near 1. Since the correlation is a continuous function, the intermediate value theorem that makes the correlation0. That value is approximately1.54..

is normally distributed one may compute its cumulative distribution function:

.

is nowhere near being normally distributed, since it has a substantial probability (about0.88) of it being equal to0. By contrast, the normal distribution, being a continuous distribution, has no discrete part are not jointly normally distributed, even though they are separately normally distributed.

Examples with support almost everywhere in the plane

of a random point in the plane are chosen according to the probability density function are uncorrelated, and each of them is normally distributed (with mean 0 and variance 1), but they are not independent.

has a Cauchy distribution. is a Chi-squared random variable with two degrees of freedom.

are uncorrelated as the bivariate distributions all have reflection symmetry across the axes.

; 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.

See also

• Correlation and dependence

References

1. Notes