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Suppose X1,..., X, i.i.d. N(H, 1). Show that p(x. 1.96 1.96 P(Xn = 0.95

Suppose X1,..., X, i.i.d. N(H, 1). Show that p(x. 1.96 1.96 P(Xn = 0.95

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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(a) Suppose X1,..., X,m i.i.d. N(µ, 1). Show that 1.96 1.96 P(X, = 0.95 2 Hint: These two facts may be helpful: (i) a linear combination of independent normal r.v.'s is normal (see DeGroot Theorem 5.6.7) (ii) if Z ~ N(0, 1), then P(-1.96 < Z < 1.96) = 0.95. (b) Suppose X1,..., X, i.i.d. F with mean µ and variance o². (F is a general distribution and not necessarily normal.) When n is large, show that P (x.- Р (Х, — 1.96- <H< Xn + 1.96– 2 0.95. Suppose X1,..., X, F with mean µ and variance o?. (F is a general (c) distribution and not necessarily normal.) For a < 1, suppose c is the (1 – a/2)-quantile of N(0, 1). When n is large, show that P( Xn - <H< Xn+ 2 1- a. Hint: Review the definition of quantile in Lecture Note 1, or from 120A. Note: When o is known and µ is considered an unknown but fixed (i.e., not random) parameter value, (Xm 1.96, X, + 1.96) is an approximate 95% confidence in- terval for µ, exactly due to the result you showed in (b). Similarly, an approrimate (1 – a) × 100% confidence interval for u can be constructed using result you showed in (c). Confidence intervals will be studied systematically later this quarter. Think about how you would interpret the probability statement you proved in (b) – we will delve deeper into this later this quarter.
Transcribed Image Text:(a) Suppose X1,..., X,m i.i.d. N(µ, 1). Show that 1.96 1.96 P(X, = 0.95 2 Hint: These two facts may be helpful: (i) a linear combination of independent normal r.v.'s is normal (see DeGroot Theorem 5.6.7) (ii) if Z ~ N(0, 1), then P(-1.96 < Z < 1.96) = 0.95. (b) Suppose X1,..., X, i.i.d. F with mean µ and variance o². (F is a general distribution and not necessarily normal.) When n is large, show that P (x.- Р (Х, — 1.96- <H< Xn + 1.96– 2 0.95. Suppose X1,..., X, F with mean µ and variance o?. (F is a general (c) distribution and not necessarily normal.) For a < 1, suppose c is the (1 – a/2)-quantile of N(0, 1). When n is large, show that P( Xn - <H< Xn+ 2 1- a. Hint: Review the definition of quantile in Lecture Note 1, or from 120A. Note: When o is known and µ is considered an unknown but fixed (i.e., not random) parameter value, (Xm 1.96, X, + 1.96) is an approximate 95% confidence in- terval for µ, exactly due to the result you showed in (b). Similarly, an approrimate (1 – a) × 100% confidence interval for u can be constructed using result you showed in (c). Confidence intervals will be studied systematically later this quarter. Think about how you would interpret the probability statement you proved in (b) – we will delve deeper into this later this quarter.
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