Jacek Wallusch _________________________________ Statistics for International Business Lecture 8: Distributions and Densities.

Slides:

Advertisements

Similar presentations

The Normal Distribution

Advertisements

Chapter 6 Continuous Random Variables and Probability Distributions

E(X 2 ) = Var (X) = E(X 2 ) – [E(X)] 2 E(X) = The Mean and Variance of a Continuous Random Variable In order to calculate the mean or expected value of.

Stat381 Cheat sheet: Confidence Intervals

Chi-Squared Distribution Leadership in Engineering

Some Continuous Probability Distributions By: Prof. Gevelyn B. Itao.

Econ 140 Lecture 41 More on Univariate Populations Lecture 4.

REVIEW t-Distribution t-Distribution. t-Distribution The student t distribution was first derived by William S. Gosset in t is used to represent.

1 The Chi squared table provides critical value for various right hand tail probabilities ‘A’. The form of the probabilities that appear in the Chi-Squared.

Statistics Lecture 16. Gamma Distribution Normal pdf is symmetric and bell-shaped Not all distributions have these properties Some pdf’s give a.

Probability and Statistics Review

The moment generating function of random variable X is given by Moment generating function.

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Chapter 7 Statistical Intervals Based on a Single Sample.

Some Continuous Probability Distributions Asmaa Yaseen.

Confidence Intervals for the Mean (σ Unknown) (Small Samples)

The table shows a random sample of 100 hikers and the area of hiking preferred. Are hiking area preference and gender independent? Hiking Preference Area.

AP STATISTICS LESSON 13 – 1 (DAY 1) CHI-SQUARE PROCEDURES TEST FOR GOODNESS OF FIT.

SECTION 6.4 Confidence Intervals for Variance and Standard Deviation Larson/Farber 4th ed 1.

Moment Generating Functions

Chapter 10 The Normal and t Distributions. The Normal Distribution A random variable Z (-∞ ∞) is said to have a standard normal distribution if its probability.

Sampling Distributions & Standard Error Lesson 7.

JMB Ch6 Lecture2 Review EGR 252 Spring 2011 Slide 1 Continuous Probability Distributions Many continuous probability distributions, including: Uniform.

H1H1 H1H1 HoHo Z = 0 Two Tailed test. Z score where 2.5% of the distribution lies in the tail: Z = Critical value for a two tailed test.

Chapter 26 Chi-Square Testing

Testing Differences in Population Variances

Section 8-5 Testing a Claim about a Mean: σ Not Known.

Section 9.3: Confidence Interval for a Population Mean.

Chapter 12 Continuous Random Variables and their Probability Distributions.

Confidence Intervals for the Mean (Small Samples) 1 Larson/Farber 4th ed.

The Chi-Square Distribution. Preliminary Idea Sum of n values of a random variable.

Inferences Concerning Variances

Ch8.2 Ch8.2 Population Mean Test Case I: A Normal Population With Known Null hypothesis: Test statistic value: Alternative Hypothesis Rejection Region.

Chapter 4 Continuous Random Variables and Probability Distributions  Probability Density Functions.2 - Cumulative Distribution Functions and E Expected.

Probability and Statistics Dr. Saeid Moloudzadeh Uniform Random Variable/ Normal Random Variable 1 Contents Descriptive Statistics Axioms.

Section 8-6 Testing a Claim about a Standard Deviation or Variance.

Section 6.2 Confidence Intervals for the Mean (Small Samples) Larson/Farber 4th ed.

1 Probability Distributions Random Variable A numerical outcome of a random experiment Can be discrete or continuous Generically, x Probability Distribution.

Chi Square Test for Goodness of Fit Determining if our sample fits the way it should be.

Two Sample Problems  Compare the responses of two treatments or compare the characteristics of 2 populations  Separate samples from each population.

Section 6.2 Confidence Intervals for the Mean (Small Samples) © 2012 Pearson Education, Inc. All rights reserved. 1 of 83.

Chapter 14 Single-Population Estimation. Population Statistics Population Statistics:  , usually unknown Using Sample Statistics to estimate population.

Jacek Wallusch _________________________________ Statistics for International Business Lecture 12: Correlation.

Chapter 8 Confidence Intervals Copyright © 2014 by The McGraw-Hill Companies, Inc. All rights reserved.McGraw-Hill/Irwin.

Theoretical distributions: the other distributions.

Statistical Inferences for Population Variances

Summary of t-Test for Testing a Single Population Mean (m)

ASV Chapters 1 - Sample Spaces and Probabilities

The Exponential and Gamma Distributions

Chapter 4 Continuous Random Variables and Probability Distributions

Point and interval estimations of parameters of the normally up-diffused sign. Concept of statistical evaluation.

Confidence Interval of the Mean – Population St. Dev Unknown

Chapter 4. Inference about Process Quality

Chapter 6 Confidence Intervals.

Math 4030 – 10b Inferences Concerning Variances: Hypothesis Testing

Active Learning Lecture Slides

Chapter 7: Sampling Distributions

t distribution Suppose Z ~ N(0,1) independent of X ~ χ2(n). Then,

Section 6-4 – Confidence Intervals for the Population Variance and Standard Deviation Estimating Population Parameters.

STAT 5372: Experimental Statistics

The Normal Probability Distribution Summary

Chapter 6 Confidence Intervals.

Distributions and Densities: Gamma-Family and Beta Distributions

8.3 – Estimating a Population Mean

Confidence Intervals for Proportions and Variances

6.3 Sampling Distributions

Central Limit Theorem: Sampling Distribution.

Distributions and Densities

Chapter 6 Confidence Intervals.

Statistical Inference for the Mean: t-test

Moments of Random Variables

Presentation transcript:

Jacek Wallusch _________________________________ Statistics for International Business Lecture 8: Distributions and Densities

Gamma Distribution ____________________________________________________________________________________________ Statistics: 8 Gamma function Data: Excel file forex.xls Properties: positive skewness and nonnegative values of the random variables Formula:

Gamma Distribution ____________________________________________________________________________________________ Statistics: 8  2 distribution k – degrees of freedom Gamma distribution: Characteristics of : Characteristics of  2 distribution : degrees of freedom – minimum number of independent random variables describing the system and influencing the result

Gamma Distribution ____________________________________________________________________________________________ Statistics: 8  2 distribution k – degrees of freedom...or in other words: then: define the sequence X define W

 2 Distribution ____________________________________________________________________________________________ Statistics: 8 generator k – degrees of freedom Excel function: =ROZKŁAD.CHI(X;df) X – value, at which the distribution should be evaluated df – number of degrees of freeedom

Gamma Distribution ____________________________________________________________________________________________ Statistics: 8  2 distribution d.f. – degrees of freedom distribution

Gamma Distribution ____________________________________________________________________________________________ Statistics: 8  2 distribution d.f. – degrees of freedom distribution

Gamma Distribution ____________________________________________________________________________________________ Statistics: 8 F distribution k – degrees of freedom define first: define a new variable: then:

Gamma Distribution ____________________________________________________________________________________________ Statistics: 8 Student t distribution k – degrees of freedom define first: define a new variable: then:

Beta Distribution ____________________________________________________________________________________________ Statistics: 8 Beta  – left parameters,  – right parameters Distribution: Examples:

Calculating Probabilities ____________________________________________________________________________________________ Statistics: 8 Excel k – degrees of freedom, T - tails Excel Functions: calculating probability for known value of random variable X =ROZKŁAD.type(X;k;T) calculating the value of random variable X for known probability value =ROZKŁAD.type.ODW(Prob; k)

Calculating Probabilities ____________________________________________________________________________________________ Statistics: 8 distribution k – degrees of freedom t-Stat: =ROZKŁAD.T(X;k;Tails) =ROZKŁAD.T.ODW(Prob;k) F-Stat: =ROZKŁAD.F(X;k 1 ;k 2 ) =ROZKŁAD.F.ODW(Prob; k 1 ;k 2 )  2 -Stat: =ROZKŁAD.CHI(X;k) =ROZKŁAD.CHI.ODW(Prob; k)