5.1.1 CHI-SQUARE DISTRIBUTION CHARACTERISTICS

The chi-square, X ², distribution is widely used in statistics. Its pdf is given by [1, p. 108]

(5.1)

where

and

Figure 5.1 shows how the shape of the X ² pdf changes as v increases. Some of the X ² pdf's characteristics are the following:

1. The mean, ? ², of the ? ² pdf is

   (5.2)

2. The standard deviation, , of the ? ² pdf is

   (5.3)

3. The coefficient of skewness, ? ₃, of the ? ² pdf is given by [2, p. 4 53]

   (5.4)

4. The coefficient of kurtosis, ? ₄, of the ? ² pdf is given by

   (5.5)

5. The ? ² pdf has a single mode at

   (5.6)

6. The additivity of ? ² random variables:

   If and are independent, ? ² random variables with v ₁ and v ₁ degrees of freedom, respectively, the sum of these random variables; i.e.,

   (5.7)

   also has a ? ² distribution with v = v ₁ + v ₂ degrees of freedom [1].

7. The relation to the normal distribution:

   Let X ₁, X ₂,..., X _n be n independent, normally distributed random variables, each with zero mean and unit variance. Then the random variable

   (5.8)

   has a ? ² distribution with