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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.

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Special Continuous Probability Distributions t- Distribution Chi-Squared Distribution

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  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

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