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Let X₁, X₂, X3,..., X be a random sample of size n from population X. Suppose that X~N(0,1) and Y = -√. i) Show that the standard score of the sample mean X, is equal to Y. ii) Show that the mean and variance of the random variable Y are 0 and 1, respectiv iii) Show using the moment generating function technique that Y is a standard normal random variable.

Let X₁, X₂, X3,..., X be a random sample of size n from population X. Suppose that X~N(0,1) and Y = -√. i) Show that the standard score of the sample mean X, is equal to Y. ii) Show that the mean and variance of the random variable Y are 0 and 1, respectiv iii) Show using the moment generating function technique that Y is a standard normal random variable.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.8: Probability
Problem 31E
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a) Let X₁, X₂, X3,... X, be a random sample of size n from population X. Suppose that X-N (0,1) and Y = -√n. i) Show that the standard score of the sample mean X, is equal to Y. ii) Show that the mean and variance of the random variable Y are 0 and 1, respectiv iii) Show using the moment generating function technique that Y is a standard normal random variable.
Transcribed Image Text:a) Let X₁, X₂, X3,... X, be a random sample of size n from population X. Suppose that X-N (0,1) and Y = -√n. i) Show that the standard score of the sample mean X, is equal to Y. ii) Show that the mean and variance of the random variable Y are 0 and 1, respectiv iii) Show using the moment generating function technique that Y is a standard normal random variable.
a) Let X₁, X₂, X3,... X, be a random sample of size n from population X. Suppose that X-N (0,1) and Y = -√n. i) Show that the standard score of the sample mean X, is equal to Y. ii) Show that the mean and variance of the random variable Y are 0 and 1, respectiv iii) Show using the moment generating function technique that Y is a standard normal random variable.
Transcribed Image Text:a) Let X₁, X₂, X3,... X, be a random sample of size n from population X. Suppose that X-N (0,1) and Y = -√n. i) Show that the standard score of the sample mean X, is equal to Y. ii) Show that the mean and variance of the random variable Y are 0 and 1, respectiv iii) Show using the moment generating function technique that Y is a standard normal random variable.
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