Suppose X and Y are independent random variables with the same mean µ but possibly different variances σx2 and σy2.Imagine that X and Y are the averages of independent samples of size n1 and n2, respectively, taken from populations with the same mean µ and the same variance σ2. Apply below to identify the α that minimizes and the minimum value. Be sure to simplify both the new estimator and the minimum variance. Interpret the result.The variance of αX+(1-α)Y, which also has mean µ, is minimized with α=σy2/(σX2+σy2).

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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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Suppose X and Y are independent random variables with the same mean µ but possibly different variances σ_x² and σ_y².

Imagine that X and Y are the averages of independent samples of size n₁ and n₂, respectively, taken from populations with the same mean µ and the same variance σ².

Apply below to identify the α that minimizes and the minimum value. Be sure to simplify both the new estimator and the minimum variance. Interpret the result.

The variance of αX+(1-α)Y, which also has mean µ, is minimized with α=σ_y²/(σ_X²+σ_y²).

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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