The x2 Chi Square Distribution

Let Zi, Z2,..., Zk be independent standardized normal variables (i.e., normal variables with zero mean and unit variance). Then the quantity k

¿=i is said to possess the x 2 distribution with k degrees of freedom (df), where the term df means the number of independent quantities in the previous sum. A chi-square-distributed variable is denoted by xi, where the subscript k indicates the df. Geometrically, the chi-square distribution appears in Figure A.5.

Properties of the x2 distribution are as follows:

1. As Figure A.5 shows, the x2 distribution is a skewed distribution, the degree of the skewness depending on the df. For comparatively few df, the distribution is highly skewed to the right; but as the number of df increases, the distribution becomes increasingly symmetrical. As a matter of fact, for df in excess of 100, the variable can be treated as a standardized normal variable, where k is the df.

2. The mean of the chi-square distribution is k, and its variance is 2k, where k is the df.

3. If Z1 and Z2 are two independent chi-square variables with k1 and k2 df, then the sum Z1 + Z2 is also a chi-square variable with df = k1 + k2.

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