sums of normal random variables need not be normal
A common misconception among students of probability theory is the belief that the sum of two normally distributed (http://planetmath.org/NormalRandomVariable) random variables is itself normally distributed. By constructing a counterexample, we show this to be false.
It is however well known that the sum of normally distributed variables will be normal under either of the following situations.
are joint normal (http://planetmath.org/JointNormalDistribution).
The statement that the sum of two independent normal random variables is itself normal is a very useful and often used property. Furthermore, when working with normal variables which are not independent, it is common to suppose that they are in fact joint normal. This can lead to the belief that this property holds always.
is zero. While it is certainly true that independent variables have zero covariance, the converse statement does not hold. Again, it is often supposed that they are joint normal, in which case a zero covariance will indeed imply independence.
We construct a pair of random variables satisfying the following.
and each have the standard normal distribution.
The covariance, , is zero.
The sum is not normally distributed, and and are not independent.
We start with a pair of independent random variables where has the standard normal distribution and takes the values , each with a probability of . Then set,
It can similarly be shown that is equal to . So, has the same distribution as and is normal with mean zero and variance one.
Using the fact that has zero mean and is independent of , it is easily shown that the covariance of and is zero.
As and have zero covariance and each have variance equal to , the sum will have variance equal to . Also, the sum satisfies
In particular, this shows that . However, normal random variables with nonzero variance always have a positive probability of being greater than any given real number. So, is not normally distributed.
This also shows that, despite having zero covariance, and are not independent. If they were, then the fact that sums of independent normals are normal would imply that is normal, contradicting what we have just demonstrated.
|Title||sums of normal random variables need not be normal|
|Date of creation||2013-03-22 18:43:44|
|Last modified on||2013-03-22 18:43:44|
|Last modified by||gel (22282)|