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

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

Calculus For The Life Sciences

2nd Edition

ISBN:9780321964038

Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Chapter1: Functions

Section1.2: The Least Square Line

Problem 5E

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Question

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).

Expert Solution