Probability and Statistical Inference

Math 224 Fall 2017 Homework 5 Drew Armstrong

Problems from 9th edition of Probability and Statistical Inference by Hogg, Tanis and Zim- merman:

• Section 3.1, Exercises 3, 10. • Section 3.3, Exercises 2, 3, 10, 11. • Section 5.6, Exercises 2, 4. • Section 5.7, Exerciese 1, 4, 14.

Additional Problems.

1. The Normal Curve. Let µ, σ2 ∈ R be any real numbers, with σ2 ≥ 0, and consider the graph of the function

n(x) = 1√

2πσ2 e−(x−µ)

2/2σ2 .

(a) Compute the first derivative n′(x) and show that n′(x) = 0 implies x = µ. (b) Compute the second derivative n′′(x) and show that n′′(µ) < 0, so that the curve has

a local maximum at x = µ. (c) Show that n′′(x) = 0 implies x = µ+σ or x = µ−σ, hence the curve has inflections at

these points. [The existence of inflections at µ+ σ and µ− σ was de Moivre’s original motivation for defining the standard deviation.]