The normal distribution has been playing an important role in the development of statistical theories. This subsection briefly describes two distributions related to the functions of normal random variables.
Figure 2.23 Shapes of standard beta probability density functions. (After Johnson and Kotz, 1972.)
|
|
X2 (chi-square) distribution. The sum of the squares of K independent standard normal random variables results in a x2 (chi-square) random variable with K degrees of freedom, denoted as x2- In other words,
E z2 ~ xK (2-101)
k=і
in which the Zks are independent standard normal random variables. The PDF of a x 2 random variable with K degrees of freedom is
f x2(* I K) = 2k/2r1 K/2)x(K/2-1)e-x/2 for x > 0 (2.102)
Comparing Eq. (2.102) with Eq. (2...
read more