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Statistics: Principles and Methods, 7th Edition by Gouri K. Bhattacharyya, Richard A. Johnson

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APPENDIX A4

The Expected Value and Standard Deviation of images

Some basic properties of the sampling distribution of images can be expressed in terms of the population mean and variance when the observations form a random sample. Let

images

In a random sample, the random variables X1, …, Xn are independent, and each has the distribution of the population. Consequently, each observation has mean μ and variance σ2, or

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Next,

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and n is a constant. Using the additivity properties of expectation and variance discussed in Appendix A3.4, we obtain

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Furthermore, taking the square root yields

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