1 / 20

Special Continuous Probability Distributions t- Distribution Chi-Squared Distribution

Systems Engineering Program. Department of Engineering Management, Information and Systems. EMIS 7370/5370 STAT 5340 : PROBABILITY AND STATISTICS FOR SCIENTISTS AND ENGINEERS. Special Continuous Probability Distributions t- Distribution Chi-Squared Distribution F- Distribution.

gudrun
Download Presentation

Special Continuous Probability Distributions t- Distribution Chi-Squared Distribution

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Systems Engineering Program Department of Engineering Management, Information and Systems EMIS 7370/5370 STAT 5340 : PROBABILITY AND STATISTICS FOR SCIENTISTS AND ENGINEERS Special Continuous Probability Distributions t- Distribution Chi-Squared Distribution F- Distribution Dr. Jerrell T. Stracener, SAE Fellow Leadership in Engineering

  2. t-Distribution

  3. The t-Distribution • Definition - A random variable T is said to have the • t-distribution with parameter , called degrees of • freedom, if its probability density function is given by: • , -  < t < where is a positive integer.

  4. The t-Distribution Remark: The distribution of T is usually called the Student-t or the t-distribution. It is customary to let tp represent the t value above which we find an area equal to p. p tp t 0 Values of T, tp,ν for which P(T > tp,ν) = p

  5. The t-distribution probability density function for various values of  -3 -2 -1 0 1 2 3

  6. Table of t-Distribution t-table gives values of tp for various values of p and ν. The areas, p, are the column headings; the degrees of freedom, ν, are given in the left column, and the table entries are the t values. • Excel

  7. t-Distribution - Example If T~t10, find: (a) P(0.542 < T < 2.359) (b) P(T < -1.812) (c) t′ for which P(T>t′) = 0.05 .

  8. Chi-Squared Distribution

  9. The Chi-Squared Distribution • Definition - A random variable X is said to have the • Chi-Squared distribution with parameter ν, called • degrees of freedom, if the probability density function • of X is: • , for x >0 • , elsewhere where ν is a positive integer.

  10. The Chi-Squared Model Remarks: The Chi-Squared distribution plays a vital role in statistical inference. It has considerable application in both methodology and theory. It is an important component of statistical hypothesis testing and estimation. The Chi-Squared distribution is a special case of the Gamma distribution, i.e., when  = ν/2 and  = 2.

  11. Mean or Expected Value • Standard Deviation The Chi-Squared Model - Properties

  12. f(x) p x The Chi-Squared Model - Properties It is customary to let 2p represent the value above which we find an area of p. This is illustrated by the shaded region below. • For tabulated values of the Chi-Squared distribution see the • Chi-Squared table, which gives values of 2p for various values • of p and ν. The areas, p, are the column headings; the degrees • of freedom, ν, are given in the left column, and the table entries • are the 2 values. • Excel

  13. The Chi-Squared Model – Example c 2 If , find: (a) P(7.261 < X < 24.996) (b) P(X < 6.262) (c) c’ for which P(X < c’) = 0.02 X ~ 15

  14. F-Distribution

  15. The F-Distribution Definition - A random variable X is said to have the F-distribution with parameters ν1 and ν2, called degrees of freedom, if the probability density function is given by: , 0 < x < , elsewhere The probability density function of the F-distribution depends not only on the two parameters ν1 and ν2 but also on the order in which we state them.

  16. The F-Distribution - Application Remark: The F-distribution is used in two-sample situations to draw inferences about the population variances. It is applied to many other types of problems in which the sample variances are involved. In fact, the F-distribution is called the variance ratio distribution.

  17. 6 and 24 d.f. 6 and 10 d.f. x 0 The F-Distribution f(x) Probability density functions for various values of ν1 and ν2

  18. The F-Distribution - Properties • Table: The fp is the f value above which we find an area • equal to p, illustrated by the shaded area below. f(x) p x • For tabulated values of the F-distribution see the F • table, which gives values of xp for various values of ν1 • and ν2. The degrees of freedom, ν1 and ν2 are the • column and row headings; and the table entries are • the x values. • Excel

  19. Let x(ν1, ν2) denote x with ν1 and ν2 degrees of freedom, then The F-Distribution - Properties

  20. The F-Distribution – Example If Y ~ F6,11, find: (a) P(Y < 3.09) (b) y’ for which P(Y > y’) = 0.01

More Related