4. seminar
Problem 1
Independent laboratory measurements of particular constant  are characterized by a
random sample X1, . . . , Xn, E(Xi) = , D(Xi) = 2
, i = 1, . . . , n. Consider statistics
Mn = 1
n
n
i=1
Xi and Ln = X1+X2
2
.
a) Prove that Mn and Ln are unbiased estimators of the constant .
b) Find out which of these estimators is better.
c) Prove that {Mn} and {Ln} make the sequence of asymptotically unbiased estimators
of the parameter .
d) Prove that {Mn} and {Ln} make the sequence of consistent estimators of the parameter
.
Problem 2
Let X11, . . . , X1n1 and X21, . . . , X2n2 be two independent random samples. The first sample
follows the distribution with expected value 1 and variance 2
, the second sample follows
the distribution with expected value 2 and variance 2
. Let M1, M2 denote sample means;
let S2
1 , S2
2 denote sample variances and S2
 =
(n1-1)S2
1 +(n2-1)S2
2
n1+n2-2
denote weighted mean of
sample variances.
a) Prove that the statistic M1 - M2 is an unbiased estimator of the parametric function
1 - 2.
b) Prove that the statistic S2
 is an unbiased estimator of the parametric function 2
.
Problem 3
Let X1, . . . , Xn be a random sample from the continuous uniform distribution U(0, b),
where b > 0 is an unknown parameter. The following statistics are defined:
T1 = X1 + 1
2
X2 + 1
3
X3 + 1
6
X4 and T2 = 1
2
(X1 + X2 + X3 + X4).
a) Show that statistics T1 and T2 are unbiased estimators of the parameter b.
b) Decide which estimator is better.
c) Suggest any other estimator.
Let X(n) = max{X1, . . . , Xn}. Show that the statistic T3 = X(n) is a consistent
estimator of parameter b.
Problem 4
The plane speed was measured 5 times and the realization of sample mean was 870,3 m/sec.
Find 95% confidence interval for parameter  if the plane speed follows normal distribution
with standard deviation  = 2.1m/s.