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(a) Let Z follow a standard normal distribution. i. Find the equi-tailed 95% probability interval, i.e. find a > 0 such that P(Z e (-a, a)) = 0.95 and express your result in terms of the inverse function -1 of the standard normal cdf (note that the inverse function satisfies -(4(x)) = x for all r € R). Finally write down the value of a to three significant digits. ii. Instead of an equi-tailed interval, consider the interval (-a, b) where a, b> 0 are such that P(Z € (-a, b)) = 0.95. Express b as a function b(a) of a. It may help to express your results in terms of -1 and . Show that the derivative of the length I of the interval, I = b(a) + a, is given by (a) $ (-(0.95 + (-a)))' dl da where o denotes the standard normal pdf. Hence obtain a candidate value of a for which the length of the interval is minimal by equating this derivative to zero. You do not need to show that this candidate value is an actual minimizer.

(a) Let Z follow a standard normal distribution. i. Find the equi-tailed 95% probability interval, i.e. find a > 0 such that P(Z e (-a, a)) = 0.95 and express your result in terms of the inverse function -1 of the standard normal cdf (note that the inverse function satisfies -(4(x)) = x for all r € R). Finally write down the value of a to three significant digits. ii. Instead of an equi-tailed interval, consider the interval (-a, b) where a, b> 0 are such that P(Z € (-a, b)) = 0.95. Express b as a function b(a) of a. It may help to express your results in terms of -1 and . Show that the derivative of the length I of the interval, I = b(a) + a, is given by (a) $ (-(0.95 + (-a)))' dl da where o denotes the standard normal pdf. Hence obtain a candidate value of a for which the length of the interval is minimal by equating this derivative to zero. You do not need to show that this candidate value is an actual minimizer.

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) Let Z follow a standard normal distribution. i. Find the equi-tailed 95% probability interval, i.e. find a > 0 such that P(Z € (-a, a)) = 0.95 and express your result in terms of the inverse function -1 of the standard normal cdf (note that the inverse function satisfies -((x)) = x for all r E R). Finally write down the value of a to three significant digits. ii. Instead of an equi-tailed interval, consider the interval (-a, b) where a, b >0 are such that P(ZE (-a, b)) = 0.95. Express b as a function b(a) of a. It may help to express your results in terms of -1 and . Show that the derivative of the length I of the interval, I = b(a) + a, is given by dl = 1 da (a) (-(0.95+ (-a))' where o denotes the standard normal pdf. Hence obtain a candidate value of a for which the length of the interval is minimal by equating this derivative to zero. You do not need to show that this candidate value is an actual minimizer.
Transcribed Image Text:(a) Let Z follow a standard normal distribution. i. Find the equi-tailed 95% probability interval, i.e. find a > 0 such that P(Z € (-a, a)) = 0.95 and express your result in terms of the inverse function -1 of the standard normal cdf (note that the inverse function satisfies -((x)) = x for all r E R). Finally write down the value of a to three significant digits. ii. Instead of an equi-tailed interval, consider the interval (-a, b) where a, b >0 are such that P(ZE (-a, b)) = 0.95. Express b as a function b(a) of a. It may help to express your results in terms of -1 and . Show that the derivative of the length I of the interval, I = b(a) + a, is given by dl = 1 da (a) (-(0.95+ (-a))' where o denotes the standard normal pdf. Hence obtain a candidate value of a for which the length of the interval is minimal by equating this derivative to zero. You do not need to show that this candidate value is an actual minimizer.
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