Answered: Using a long rod that has length m, you…

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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Question

Using a long rod that has length m, you are going
to lay out a square plot in which the length of each
side is m. Thus the area of the plot will be m2
.
However, you do not know the value of m, so
you decide to make n independent measurements
X1, X2,... Xn of the length. Assume that each Xi
has mean m (unbiased measurements) and variance s2
.
a. Show that X2 is not an unbiased estimator
for m2
. [Hint: For any rv Y, E(Y2
) ¼
V(Y) + [E(Y)]2
. Apply this with Y ¼ X.]
b. For what value of k is the estimator X2 - kS2
unbiased for m2
? [Hint: Compute E(X2 - kS2).]

Definition Definition Measure of central tendency that is the average of a given data set. The mean value is evaluated as the quotient of the sum of all observations by the sample size. The mean, in contrast to a median, is affected by extreme values. Very large or very small values can distract the mean from the center of the data. Arithmetic mean: The most common type of mean is the arithmetic mean. It is evaluated using the formula: μ = 1 N ∑ i = 1 N x i Other types of means are the geometric mean, logarithmic mean, and harmonic mean. Geometric mean: The nth root of the product of n observations from a data set is defined as the geometric mean of the set: G = x 1 x 2 ... x n n Logarithmic mean: The difference of the natural logarithms of the two numbers, divided by the difference between the numbers is the logarithmic mean of the two numbers. The logarithmic mean is used particularly in heat transfer and mass transfer. ln x 2 − ln x 1 x 2 − x 1 Harmonic mean: The inverse of the arithmetic mean of the inverses of all the numbers in a data set is the harmonic mean of the data. 1 1 x 1 + 1 x 2 + ...

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