MA22S6 Numerical and Data Analysis 1 2015-2016
Homework sheet 6
(Due Thursday, March 24 in class)
1. We consider a set of n independent random variables X1 . . . Xn with expected values µ1 , µ2 , . . . , µn
and variances σ12 , σ22 , . . . , σn2 . This set constitutes a random sample if µ1 = µ2 = . . . = µn ≡ µ
and σ12 = σ22 = . . . = σn2 ≡ σ 2 .
a) Show that, for a random sample, the sample mean
X̄n =
n
1X
Xi
n i=1
is an unbiased estimator of the expected value µ . (Unbiased means that its expected value
is equal to µ even for finite n, not just in the limit of n → ∞.).
b) Compute the variance of this estimator.
c) Show that
n
1X
(Xi − X̄n )2
n i=1
is a biased estimator of the variance σ 2 and give an unbiased estimator.
2. Explain why fitting to a constant is the same as computing a weighted average.
3. The fuel consumption (in litres per 100
of the average speed.
i
1
2
3
km) of a certain type of car is measured as a function
xi
yi
σi
40 4.45 0.45
80 9.65 0.25
100 10.50 0.30
a) Write a simple model under the assumption that the fuel consumption is directly proportional to the average speed.
b) Use the χ2 statistic to compute the best value(s) of the model parameter(s) and give a
one-sigma confidence level interval.
c) Comment on the quality of the model and compare the model with the data.
4. We measure a quantity y with sampling error σ
measurement i = 1, . . . 4, we find
yi
i xi
1 3 15.2
2 7 17.8
3 10 21.5
4 16 27.4
as a function of a parameter x. For each
σi
0.10
0.16
0.29
0.23
We want to model the data by y = αx + β.
a) Is this a priori a reasonable model ? Justify your answer by a graph.
b) Use the χ2 statistic to compute the best values of the model parameters and give a onesigma confidence level interval.
c) Comment on the quality of the model and compare the model with the data.
1
Lecturer: Stefan Sint, sint@maths.tcd.ie