(a) Find the points of inflection of the curve of (1). (b) Considering Φ 2 () and introducing polar coordinates in the double integral (a standard trick worth remembering), prove (c) Show that Ï in (1) is indeed the standard deviation of the normal distribution. (d) Bernoullis law of large numbers. In an experiment let an event A have probability

Chapter 24, PROBLEM SET 24.8 #14
(a) Find the points of inflection of the curve of (1).

(b) Considering Φ2(ˆž) and introducing polar coordinates in the double integral (a standard trick worth remembering), prove

-и? (2 ди — du = 1. (12) Ф(с) %3D Ф(о) V2п

(c) Show that σ in (1) is indeed the standard deviation of the normal distribution. 

(d) Bernoulli€™s law of large numbers. In an experiment let an event A have probability p(0 < p < 1) and let X be the number of times A happens in n independent trials. Show that for any given ˆˆ > 0,

+(E-|=-)-1 as n→ 00.

(e) If X is normal with mean μ and variance σ2, show that X* = c1X + c2 (c1 > 0) is normal with mean μ* = c1μ + c2 and variance σ*2 = c21σ2. 

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Related Book For answer-question

Advanced Engineering Mathematics

10th edition

Authors: Erwin Kreyszig

ISBN: 978-0470458365