## E. Useful random variables

#### Normal: A random variable X is normally distributed if its density function is exactly of the form:

where is the mean of X and is the variance of X.
We indicate that X is a normal random variable by writing: X ~ N( , )

#### 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 of freedom. We write or sometimes .

#### t-distribution or Let Z ~ N(0,1) and C ~ where Z and C are indendent. Then the function

follows the distribution.

#### Important properties of the normal distribution

1. A linear combination of normal random variables is also normally distributed. Let be m normally distributed random variables, and let be m constants. Then the random variable

is also normally distributed.
2. If X and Y are both normal then Cov[X,Y] = 0 X and Y independent.

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