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FACOLTA' DI ECONOMIA - UNIVERSITA' DI BOLOGNA
STATISTICS LMEC 2008
LIKELIHOOD FUNCTION AND MAXIMUM LIKELIHOOD ESTIMATION
Ex 1 (Poisson)
Given a random sample of n independent observation of a random quantity X distributed as
Poisson with parameter ,
a)
Write the likelihood function
b)
Find the max likelihood estimator of 
If the following data are observed:
x1 = 0, x2 = 2, x3 =3, x4 = 2, x5 = 1, x6 = 1, x7 = 1, x8 = 2, x9 = 4, x10 = 2, x11 = 0, x12 = 3.
c)
d)
Find the max likelihood estimate of the mean and the standard deviation.
Find the max likelihood estimate of Pr(X = 0) and of Pr(X = 1).
Ex 2 (coin flipping).
Find the max likelihood estimator of the parameter p for n independent Bernoulli trials with
probability p of success in each trial.
Ex 3
In the same case as before, find the max likelihood estimate of p when n = 12 and the
number of successes observed is 9, if the only possible values of p are {0.2, 0.45, 0.6, 0.75, 0.90}
Ex 4
Find the maximum likelihood estimator of parameter p for the GEOMETRIC distribution:
Pr(X = k) = p(1 - p)k-1
k = 1,2,...
Ex 5
The time to failure of an electrical bulb is a random variable X with probability distribution
depending on EX
P(X < x) = F(x) = 1 - e-x/
x>0
To check on the quality of the bulbs, a sample of 10 bulbs, chosen at random, is observed and the
following failure times (in hours) are recorded
157, 143, 188, 206, 197, 234, 189, 179, 212, 217
On the basis of these data
a) Write the likelihood function
b) Estimate by maximum likelihood.
Ex 6 (normal distribution)
Given a r.v. with normal distribution, unknown mean and known variance,
a)
find the MLE of the mean, based on a random sample of size n.
b)
What if the mean is known and the variance unknown?
c)
What if they are both unknown?
Ex 7
Let T be a random waiting time whose distribution has Rayleigh density, namely
fT(t) = (2t/2)exp{-(t/)2}
=0
if t > 0,
elsewhere.
c) Write the likelihood function given a sample of n independent observations
d) Find the maximum likelihood estimator of .
Ex 8
Let the cdf of a r.v. X be:
= 0
F(x) = x2/2
=1
if
if
if
x <0
0 ≤ x ≤ 
x>
where  > 0 is an unknown parameter.
a)
Find the density.
b)
Write the likelihood function given a sample of n i.i.d. observations.
c)
Find the maximum likelihood estimator of 
Ex 9
The percentage of voters in a town with less than 10.000 inhabitants is a r.v. X whose
probability distribution has the following density (where > 0).
f(x) = x-1
a)
b)
c)
for 0 ≤ x ≤1,
f(x) = 0
elsewhere.
Find the mean and variance of X.
Write the likelihood function given a sample of n independent towns.
Find the maximum likelihood estimator for the mean.