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Let X1,... Xn be independent random variables, all having he same distribution with expected value u and variance o?. The random variable X, defined as the arithmetic average of these variables, is called che sample mean. That is, the sample mean is given by Li=1 X; n (a) Show that E[X] = (b) Show that Var[X] = o² /n. The random variable S2, defined by EL(Xi – X)² п - 1 s the sample variance. (Denominator is n – 1, not n, due to (d).) (c) Show that E"1(Xi - X)² = ", X? – nX. %3D

Let X1,... Xn be independent random variables, all having he same distribution with expected value u and variance o?. The random variable X, defined as the arithmetic average of these variables, is called che sample mean. That is, the sample mean is given by Li=1 X; n (a) Show that E[X] = (b) Show that Var[X] = o² /n. The random variable S2, defined by EL(Xi – X)² п - 1 s the sample variance. (Denominator is n – 1, not n, due to (d).) (c) Show that E"1(Xi - X)² = ", X? – nX. %3D

Calculus For The Life Sciences
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.
Chapter13: Probability And Calculus
Section13.CR: Chapter 13 Review
Problem 6CR
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Let X1,... Xn be independent random variables, all having the same distribution with expected value u and variance o?. The random variable X, defined as the arithmetic average of these variables, is called the sample mean. That is, the sample mean is given by (a) Show that E[X] = µ. (b) Show that Var[X] = o²/n. The random variable S2, defined by EL (Xi – X)² п — 1 is the sample variance. (Denominator is n – 1, not n, due to (d).) (c) Show that (Xi – X)? = E-, X? – nX.
Transcribed Image Text:Let X1,... Xn be independent random variables, all having the same distribution with expected value u and variance o?. The random variable X, defined as the arithmetic average of these variables, is called the sample mean. That is, the sample mean is given by (a) Show that E[X] = µ. (b) Show that Var[X] = o²/n. The random variable S2, defined by EL (Xi – X)² п — 1 is the sample variance. (Denominator is n – 1, not n, due to (d).) (c) Show that (Xi – X)? = E-, X? – nX.
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