Figure 11-12. The gamma distribution for selected values of αand θ
Chi-Square χ^2 Distribution
The chi-square probability distribution has the form
f(x) =
1
2 ν/^2 Γ(ν/2)x(ν−2)/^2 e−x/^2 , for x > 0
0 , elsewhere
(11 .74)where Γ( ) represents the gamma function^2 and ν= 1 , 2 , 3 ,.. .is a parameter called the
number of degrees of freedom. Note that the chi-square distribution is sometimes
written as the χ^2 -distribution. It is a special case of the gamma distribution when
the parameters of the gamma distribution take on the values α=ν/ 2 and θ= 2. This
distribution has the mean μ=νand variance σ^2 = 2ν.
The chi-square distribution is used in testing of hypothesis, determining confi-
dence intervals and testing differences in various statistics associated with indepen-
dent samples. The tables 11.6(a) and (b) give values for areas under the probability
density function.
(^2) Recall the gamma function is defined Γ(x) = ∫∞
0 e
−ttx− (^1) dt with the property Γ(x+ 1) = xΓ(x).