Statistics II
30 October 2015
Exercise 1 (5 points)
A random sample of size n = 18 is obtained from a normally distributed population with a population mean of
μ = 46 and a variance of σ2 = 50.
(a) What is the probability that the sample mean is greater than 50?
(b) What is the value of the sample quasivariance such that 5% of the sample variances would be less
than this value?
(From: Newbold, P., W. Carlson and B. Thorne. Statistics for Business and Economics. Pearson-Prentice Hall.)
(a)
P(¯
X>50)=P
(
¯
X−μ
√
σ2
n
>50−μ
√
σ2
n
)
=P
(
T>50−46
√
50
18
)
=P(T>2.4)=1−P(T≤2.4 )=1−0.9918=0.0082
(b) For the sample quasivariance
0.05 =P
(
S2<a
)
=P
(
(n−1)S2
σ2<(n−1)a
σ2
)
=P
(
T<(18−1)a
50
)
→
0.95 =P
(
T≥(18−1)a
50
)
→
(18−1)a
50 =8.67
→
a=8.67⋅50
17 =25.5
For the sample variance
0.05 =P
(
s2<a
)
=P
(
n s2
σ2<n a
σ2
)
=P
(
T<18 a
50
)
→
0.95 =P
(
T≥18 a
50
)
→
18 a
50 =8.67
→
a=8.67⋅50
18 =24.08
Exercise 2 (5 points)
Let X be a random variable with probability function
f(x ; θ) = x
θ2e
−x2
2θ2, x ≥0, (θ> 0)
such that
E(X)=θ
√
π
2
and
Var(X)= 4−π
2θ2.
Let X = (X1,...,Xn) be a simple random sample. The
application of the method of the moments provides the estimator
^
θM=
√
2
π¯
X .
For this estimator, calculate
the bias and the mean square error, and study the consistency.
Cultural note: In probability theory and statistics, the Rayleigh distribution is a continuous probability distribution for positive-valued random
variables. A Rayleigh distribution is often observed when the overall magnitude of a vector is related to its directional components. One example
where the Rayleigh distribution naturally arises is when wind velocity is analyzed into its orthogonal 2-dimensional vector components. Assuming
that the magnitudes of each component are uncorrelated, normally distributed with equal variance, and zero mean, then the overall wind speed
(vector magnitude) will be characterized by a Rayleigh distribution. A second example of the distribution arises in the case of random complex
numbers whose real and imaginary components are i.i.d. (independently and identically distributed) Gaussian with equal variance and zero mean.
In that case, the absolute value of the complex number is Rayleigh-distributed. The distribution is named after Lord Rayleigh. (From: Wikipedia.)
Bachelor's Degree in Economics 1 Complutense University of Madrid
