Misconceptions about the normal distribution
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.
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 normal distribution.
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.
has a normal distribution with expected value 0 and variance 1.
are uncorrelated, as can be verified by calculating their covariance.
Finally, the distribution of the simple linear combination
concentrates positive probability at 0:
are not jointly normally distributed (by the definition above).
has a normal distribution with expected value 0 and variance 1.
Since the correlation is a continuous function of
, the intermediate value theorem implies there is some particular value of
—one may compute its cumulative distribution function:[6]
where the next-to-last equality follows from the symmetry of the distribution of
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.
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.
of two independent standard normal random deviates
[8][9][7]: 122 One can equally well start with the Cauchy random variable
is a Chi-squared random variable with two degrees of freedom.
results in dependences across indices: neither
are uncorrelated as the bivariate distributions all have reflection symmetry across the axes.
[citation needed] 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
The right panel shows the joint distribution of
; 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.
