A random variable X is normally distributed if its density
function is exactly of the form:
the mean of X and
variance of X.
We indicate that X is a normal random variable by
writing: X ~ N(
- Standard normal random variable (usually called Z)
is the special case Z ~ N(0,1)
- Chi-square distribution or
Let Z1, Z2, ... , Zk be k independent standard
normal random variables. Then the function
follows the chi-squared distribution with k degrees
- t-distribution or
Let Z ~ N(0,1) and C ~
where Z and C
are indendent. Then the function
- F distribution or
be indpendently distributed.
Then the function
follows the follows the
Important properties of the normal distribution
- A linear combination of normal random variables is also
normally distributed. Let
normally distributed random variables, and let
m constants. Then the random variable
is also normally distributed.
- If X and Y are both normal then
Cov[X,Y] = 0
X and Y independent.
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Document Last revised: 1997/1/5