Mathematical Methods for Physics and Engineering : A Comprehensive Guide

STATISTICS

Using (31.113) to substitute for ̄x−μ 0 in (31.116), and noting thatdx ̄=

(s/

√

N−1)dt, we find

P( ̄x, s|H 0 )d ̄xds=AsN−^1 exp

[

−

Ns^2

2 σ^2

(

1+

t^2

N− 1

)]

dt ds,

whereAis another normalisation constant. In order to obtain the sampling

distribution oftalone, we must integrateP(t, s|H 0 ) with respect tosover its

allowed range, from 0 to∞. Thus, the required distribution oftalone is given by

P(t|H 0 )=

∫∞

0

P(t, s|H 0 )ds=A

∫∞

0

sN−^1 exp

[

−

Ns^2

2 σ^2

(

1+

t^2

N− 1

)]

ds.

(31.117)

To carry out this integration, we sety=s{1+[t^2 /(N−1)]}^1 /^2 , which on substi-

tution into (31.117) yields

P(t|H 0 )=A

(

1+

t^2

N− 1

)−N/ 2 ∫∞

0

yN−^1 exp

(

−

Ny^2

2 σ^2

)

dy.

Since the integral overydoes not depend ont, it is simply a constant. We thus

find that that the sampling distribution of the variabletis

P(t|H 0 )=

1

√

(N−1)π

Γ

( 1

2 N

)

Γ

( 1

2 (N−1)

)

(

1+

t^2

N− 1

)−N/ 2

,

(31.118)

wherewehaveusedthecondition

∫∞
−∞P(t|H^0 )dt= 1 to determine the normali-
sation constant (see exercise 31.18).

The distribution (31.118) is calledStudent’st-distribution withN− 1 degrees of

freedom. A plot of Student’st-distribution is shown in figure 31.11 for various

values ofN. For comparison, we also plot the standard Gaussian distribution,

to which thet-distribution tends for largeN. As is clear from the figure, the

t-distribution is symmetric aboutt= 0. In table 31.3 we list some critical points

of the cumulative probability functionCn(t)ofthet-distribution, which is defined

by

Cn(t)=

∫t

−∞

P(t′|H 0 )dt′,

wheren=N−1 is the number of degrees of freedom. Clearly,Cn(t) is analogous

to the cumulative probability function Φ(z) of the Gaussian distribution, discussed

in subsection 30.9.1. For comparison purposes, we also list the critical points of

Φ(z), which corresponds to thet-distribution forN=∞.