Exercise 3. For an i.i.d. sample of n = 2m + 1 observations of a U(0, 1) random variable, show that the variance of the sample median is: 1 Varſi] = (17) %3D 8m + 12
Exercise 3. For an i.i.d. sample of n = 2m + 1 observations of a U(0, 1) random variable, show that the variance of the sample median is: 1 Varſi] = (17) %3D 8m + 12
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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Exercise 3
![3
The Beta Distribution and Order Statistics
Consider the special case for which X1, X2,,Xn-1, Xn is an i.i.d. sample of
a random variable X which is uniformly distributed on [0, 1].
Comparing Equation(10) for this special case, and the form of the Beta distri-
bution given by Equation (12), one easily confirms that fk is the Beta density
with a = k and B = n - k + 1.
Hence using the Theorems of the last section one obtains as a special case:
E[Y] =
k
n+1
EY?] =
k(k+1)
(n+1)(n+2)
• Var[Yµ] =
k(n-k+1)
(n+1)²(n+2)
Some interesting things to note: there's a certain symmetry inherited from the
symmetry of the uniform distribution:
Var[Y&] = Var[Yn-k]
(16)
Moreover, the variances of the order statistics are smallest at the ends and in-
crease towards the middle. This should make intuitive good sense: the minimum
and maximum of the sample are squeezed hard by their brethren observations
and have hardly any room to wiggle, and so their variance is small. The closer
to the middle an observation is, the more freedom it has to move around.
The case for which n = 2m + 1, and k = m +1 is the case for which Y is the
sample median. Note EY] = m+1
median is an unbiased estimator of the population median. ( Recall that the
population median is defined as the value u so that P(X < i) = }.)
so, unsurpisingly perhaps, the sample
%3D
Exercise 3. For an i.i.d. sample of n = 2m + 1 observations of a U(0, 1)
random variable, show that the variance of the sample median is:
1
Varſi] =
(17)
8m + 12](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F718b1378-40e4-4c32-83bc-211fc46d7de4%2Ff70149cf-7075-4a3e-8bd9-46b714568ccb%2F3iqxb5a_processed.jpeg&w=3840&q=75)
Transcribed Image Text:3
The Beta Distribution and Order Statistics
Consider the special case for which X1, X2,,Xn-1, Xn is an i.i.d. sample of
a random variable X which is uniformly distributed on [0, 1].
Comparing Equation(10) for this special case, and the form of the Beta distri-
bution given by Equation (12), one easily confirms that fk is the Beta density
with a = k and B = n - k + 1.
Hence using the Theorems of the last section one obtains as a special case:
E[Y] =
k
n+1
EY?] =
k(k+1)
(n+1)(n+2)
• Var[Yµ] =
k(n-k+1)
(n+1)²(n+2)
Some interesting things to note: there's a certain symmetry inherited from the
symmetry of the uniform distribution:
Var[Y&] = Var[Yn-k]
(16)
Moreover, the variances of the order statistics are smallest at the ends and in-
crease towards the middle. This should make intuitive good sense: the minimum
and maximum of the sample are squeezed hard by their brethren observations
and have hardly any room to wiggle, and so their variance is small. The closer
to the middle an observation is, the more freedom it has to move around.
The case for which n = 2m + 1, and k = m +1 is the case for which Y is the
sample median. Note EY] = m+1
median is an unbiased estimator of the population median. ( Recall that the
population median is defined as the value u so that P(X < i) = }.)
so, unsurpisingly perhaps, the sample
%3D
Exercise 3. For an i.i.d. sample of n = 2m + 1 observations of a U(0, 1)
random variable, show that the variance of the sample median is:
1
Varſi] =
(17)
8m + 12
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