Warning: foreach() argument must be of type array|object, bool given in /var/www/html/web/app/themes/studypress-core-theme/template-parts/header/mobile-offcanvas.php on line 20

Show that the graph of a pdf \(N\left(\mu, \sigma^{2}\right)\) has points of inflection at \(x=\mu-\sigma\) and \(x=\mu+\sigma\).

Short Answer

Expert verified
The points of inflection for the graph of a pdf of a Normal Distribution are at \(x=\mu-\sigma\) and \(x=\mu+\sigma\).

Step by step solution

01

Write out the formula for the pdf of a Normal Distribution.

The pdf of a Normal Distribution \(N\left(\mu, \sigma^{2}\right)\) is given by the formula: \[f(x)=\frac{1}{\sigma \sqrt{2 \pi}} e^{\frac{-(x-\mu)^{2}}{2\sigma^{2}}}\].
02

Take the first derivative.

Differentiate the function w.r.t. \(x\) to obtain the first derivative. Apply the chain rule to get: \[f'(x)=-\frac{(x-\mu)}{\sigma^{3} \sqrt{2 \pi}} e^{\frac{-(x-\mu)^{2}}{2\sigma^{2}}}\].
03

Take the second derivative.

Now differentiate the first derivative \(f'(x)\) w.r.t. \(x\) to obtain the second derivative. Use the chain rule for differentiation. This gives: \[f''(x)=\frac{1}{\sigma^{3} \sqrt{2 \pi}}\left(1-\frac{(x-\mu)^{2}}{\sigma^{2}}\right)e^{\frac{-(x-\mu)^{2}}{2\sigma^{2}}}\].
04

Find points of inflection.

The points of inflection are found by setting the second derivative equal to zero and solving for \(x\). This yields two solutions: \(x=\mu-\sigma\) and \(x=\mu+\sigma\). So the points of inflection are at \(x=\mu-\sigma\) and \(x=\mu+\sigma\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

See all solutions

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free