Mathematical Methods for Physics and Engineering : A Comprehensive Guide

STATISTICS

31.10 This exercise is intended to illustrate the dangers of applying formalised estimator
techniques to distributions that are not well behaved in a statistical sense.
The following are five sets of 10 values, all drawn from the same Cauchy
distribution with parametera.

(i) 4. 81 − 1. 24 1. 30 − 0. 23 2. 98

− 1. 13 − 8. 32 2. 62 − 0. 79 − 2. 85

(ii) 0. 07 1. 54 0. 38 − 2. 76 − 8. 82

1. 86 − 4. 75 4. 81 1. 14 − 0. 66

(iii) 0. 72 4. 57 0. 86 − 3. 86 0. 30

− 2. 00 2. 65 − 17. 44 − 2. 26 − 8. 83

(iv) − 0 .15 202. 76 − 0. 21 − 0. 58 − 0. 14

0. 36 0. 44 3. 36 − 2. 96 5. 51

(v) 0. 24 − 3. 33 − 1. 30 3. 05 3. 99

1. 59 − 7. 76 0. 91 2. 80 − 6. 46

Ignoring the fact that the Cauchy distribution does not have a finite variance (or

even a formal mean), show thataˆ,theMLestimatorofa, has to satisfy

s(aˆ)=

∑^10

i=1

1

1+x^2 i/aˆ^2

=5. (∗)

Using a programmable calculator, spreadsheet or computer, find the value of
aˆthat satisfies (∗) for each of the data sets and compare it with the value
a=1.6 used to generate the data. Form an opinion regarding the variance of the
estimator.
Show further that if it is assumed that(E[aˆ])^2 =E[ˆa^2 ], thenE[aˆ]=ν^12 /^2 ,where
ν 2 is the second (central) moment of the distribution, which for the Cauchy
distribution is infinite!
31.11 According to a particular theory, two dimensionless quantitiesXandY have
equal values. Nine measurements ofXgave values of 22, 11, 19, 19, 14, 27, 8,
24 and 18, whilst seven measured values ofYwere 11, 14, 17, 14, 19, 16 and

14. Assuming that the measurements of both quantities are Gaussian distributed
    with a common variance, are they consistent with the theory? An alternative
    theory predicts thatY^2 =π^2 X; are the data consistent with this proposal?
    31.12 On a certain (testing) steeplechase course there are 12 fences to be jumped, and
    any horse that falls is not allowed to continue in the race. In a season of racing
    a total of 500 horses started the course and the following numbers fell at each
    fence:
    Fence:123456789101112
    Falls: 627549293325301719111512

Use this data to determine the overall probability of a horse’s falling at a fence,

and test the hypothesis that it is the same for all horses and fences as follows.

(a) Draw up a table of the expected number of falls at each fence on the basis

of the hypothesis.

(b) Consider for each fenceithe standardised variable

zi=

estimated falls−actual falls

standard deviation of estimated falls

,

and use it in an appropriateχ^2 test.

(c) Show that the data indicates that the odds against all fences being equally

testing are about 40 to 1. Identify the fences that are significantly easier or

harder than the average.