SOME SPECIFIC PROBABILITY DISTRIBUTIONS 3.3. Moments of student's t-distribution (t()=0 (22) 4 THE F (F ISHER VARIANCE RATIO) STATISTIC 4.1. Distribution Function. If x21(v1)and x22(v2) are independently distributed chi-square vari- ates. then X(2 x2(2) has the F density with v1 and v2 degrees of freedom 4.2. Probability Density Function. The density of the F distribution is f (F; v1, v2) F>0 (24) 0 otherwise Tabulations of the distribution of F(v1, va) are widely available. Note that Fn, a(Fra m and therefore the critical values can be found from fa vi, vz The following diagram shows the pdf and cdf for the f distribution with FIGURE 8. F Distributtion pdf and cdf Probability Density Function Cumulative Distribution Function 06 2 2 Here is the pdf of the F distribution for some alternative values of pairs of values(vn and v2)SOME SPECIFIC PROBABILITY DISTRIBUTIONS 9 3.3. Moments of Student’s t-distribution. M ean (t(ν)) = 0 V ar (t(ν)) = ν ν − 2 (22) 4. The F (Fisher variance ratio) statistic 4.1. Distribution Function. If χ2 1(ν1) and χ2 2(ν2) are independently distributed chi-square vari￾ates, then F(ν1, ν2 ) = χ2 1(ν1) ν1 χ2 2(ν2) ν2 = ν2 ν1 · χ2 1(ν1) χ2 2(ν2) (23) has the F density with ν1 and ν2 degrees of freedom. 4.2. Probability Density Function. The density of the F distribution is f ( F; ν1, ν2) = Γ ( ν1+ν2 2 ) Γ ( ν1 2 ) Γ ( ν2 2 ) · ν1 ν2  ν1 2 · F ν1 2 −1 ·  1 + ν1 ν2 F  −(ν1+ν2) 2 F > 0 = 0 otherwise (24) Tabulations of the distribution of F(ν1,ν2) are widely available. Note that Fν1, ν2 ∼  1 F ν2, ν1  and therefore the critical values can be found from fα ν1 , ν2 =  1 f1−α ν2, ν1  . The following diagram shows the pdf and cdf for the F distribution with parameters ν1 = 12 and ν2 = 20. Figure 8. F Distributtion pdf and cdf 0 2 4 6 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Probability Density Function X f(X) 0 2 4 6 0 0.2 0.4 0.6 0.8 1 Cumulative Distribution Function X F(X) Here is the pdf of the F distribution for some alternative values of pairs of values (ν1 and ν2)