/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 8 Compute the measures of skewness... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 8

Compute the measures of skewness and kurtosis of a gamma distribution which has parameters \(\alpha\) and \(\beta\).

Short Answer

Expert verified
Skewness (\( \gamma_1 \)) and kurtosis (\( \gamma_2 \)) are calculated from the given parameters of the gamma distribution using the formulas: Skewness = \(2/ \sqrt{ \alpha } \) and Kurtosis = \(6/ \alpha \).

Step by step solution

01

Formula for Skewness

The formula for the skewness of a gamma distribution is: \(2/ \sqrt{ \alpha } \). Substitute \( \alpha \) with the given parameter.
02

Calculation of Skewness

Upon substitution, the skewness (\( \gamma_1 \)) is calculated based on the given parameter \( \alpha \). The computed value is the skewness of the gamma distribution.
03

Formula for Kurtosis

The formula for the kurtosis of a gamma distribution is: \(6/ \alpha \). Substitute \( \alpha \) with the given parameter.
04

Calculation of Kurtosis

Upon substitution, the kurtosis (\( \gamma_2 \)) is calculated based on the given parameter \( \alpha \). The computed value is the kurtosis of the gamma distribution and is the measure of the 'tailedness' of the probability distribution.

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 91影视!

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

Show that the moment generating function of the negative binomial distribution is \(M(t)=p^{r}\left[1-(1-p) e^{t}\right]^{-r}\). Find the mean and the variance of this distribution. Hint: In the summation representing \(M(t)\), make use of the MacLaurin's series for \((1-w)^{-r}\)

. Find the mean and variance of the \(\beta\) distribution. Hint: From the pdf, we know that $$ \int_{0}^{1} y^{\alpha-1}(1-y)^{\beta-1} d y=\frac{\Gamma(\alpha) \Gamma(\beta)}{\Gamma(\alpha+\beta)} $$ for all \(\alpha>0, \beta>0\)

Find the uniform distribution of the continuous type on the interval \((b, c)\) that has the same mean and the same variance as those of a chi-square distribution with 8 degrees of freedom. That is, find \(b\) and \(c\).

The mgf of a random variable \(X\) is \(\left(\frac{2}{3}+\frac{1}{3} e^{t}\right)^{9} .\) Show that $$ P(\mu-2 \sigma

Show that the failure rate (hazard function) of the Pareto distribution is $$ \frac{h(x)}{1-H(x)}=\frac{\alpha}{\beta^{-1}+x} $$ Find the failure rate (hazard function) of the Burr distribution with cdf $$ G(y)=1-\left(\frac{1}{1+\beta y^{\tau}}\right)^{\alpha}, \quad 0 \leq y<\infty . $$ In each of these two failure rates, note what happens as the value of the variable increases.
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Mechanics Maths

Read Explanation

Applied Mathematics

Read Explanation

Decision Maths

Read Explanation

Geometry

Read Explanation

Logic and Functions

Read Explanation
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.