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, respectively. iii) Show using the moment generating function technique that Y is a standard normal random variable. iv) What is the probability that y² is between 0.02 and 5.02?

Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.8: Probability
Problem 32E
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Question
a) Let X₁, X₂, X3,..., X₁, 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, respectively.
iii) Show using the moment generating function technique that Y is a standard normal random
variable.
iv) What is the probability that Y² is between 0.02 and 5.02?
Transcribed Image Text:a) Let X₁, X₂, X3,..., X₁, 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, respectively. iii) Show using the moment generating function technique that Y is a standard normal random variable. iv) What is the probability that Y² is between 0.02 and 5.02?
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