Mathematical Methods for Physics and Engineering - Matematica.NET

STATISTICSUsing (31.113) to substitute for ¯x − µ 0 in (31.116), and noting that d¯x =(s/ √ N − 1) dt, we find( )]P (¯x, s|H 0 ) d¯xds= As N−1 exp[− Ns22σ 2 1+ t2dt ds,N − 1where A is another normalisation constant. In order to obtain the samplingdistribution of t alone, we must integrate P (t, s|H 0 ) with respect to s over itsallowed range, from 0 to ∞. Thus, the required distribution of t alone is given by∫ ∞∫ ∞( )]P (t|H 0 )= P (t, s|H 0 ) ds = A s N−1 exp[− Ns2002σ 2 1+ t2ds.N − 1(31.117)To carry out this integration, we set y = s{1+[t 2 /(N − 1)]} 1/2 , which on substitutioninto (31.117) yields) −N/2 ∫ ∞)P (t|H 0 )=A(1+ t2y N−1 exp(− Ny2N − 1 02σ 2 dy.Since the integral over y does not depend on t, it is simply a constant. We thusfind that that the sampling distribution of the variable t is1 Γ ( 12P (t|H 0 )= √ N) ((N − 1)π Γ ( 12 (N − 1))1+ t2N − 1) −N/2,(31.118)wherewehaveusedthecondition ∫ ∞−∞ P (t|H 0) dt = 1 to determine the normalisationconstant (see exercise 31.18).The distribution (31.118) is called Student’s t-distribution with N − 1 degrees offreedom. A plot of Student’s t-distribution is shown in figure 31.11 for variousvalues of N. For comparison, we also plot the standard Gaussian distribution,to which the t-distribution tends for large N. As is clear from the figure, thet-distribution is symmetric about t = 0. In table 31.3 we list some critical pointsof the cumulative probability function C n (t) ofthet-distribution, which is definedby∫ tC n (t) = P (t ′ |H 0 ) dt ′ ,−∞where n = N − 1 is the number of degrees of freedom. Clearly, C n (t) is analogousto the cumulative probability function Φ(z) of the Gaussian distribution, discussedin subsection 30.9.1. For comparison purposes, we also list the critical points ofΦ(z), which corresponds to the t-distribution for N = ∞.1286