PROBABILITY AND STATISTICS FOR SCIENTISTS AND ENGINEERS

1. PROBABILITY AND STATISTICS FOR SCIENTISTS AND ENGINEERS Special Continuous Probability Distributionst- DistributionChi-Squared Distribution F- Distribution

2. t - DISTRIBUTION

3. t-Distribution – Probability Density Function 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. t-Distribution – Table of Probabilities 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. Values of T, tp,ν for which P(T > tp,ν) = p p tp t 0

5. -3 -2 -1 0 1 2 3 t-distribution - Probability Density Function for various values of

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.

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. Example Solution P(0.542 < T < 2.359) = 0.3-0.02 =0.28 P(T < -1.812)=F(-1.812) =P(T > 1.812)=0.05 (c) t′ for which P(T>t′) = 1-F(t′ ) =0.05 . t′ = 1.812 t 0.542 0 2.359 t -1.812 1.812 0 0.05 t t′ 0

9. Chi-Squared Distribution

10. Chi-Squared Distribution – Probability Density Function 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.

11. Chi-Squared Distribution - 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.

12. Chi-Squared Distribution – Mean and Standard Deviation Mean or Expected Value Standard Deviation

13. f(x) p x Chi-Squared Distribution – Table of Probabilities It is customary to let 2prepresent 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 2pfor 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 2values.

14. Chi-Squared Table

15. Chi-Squared Table Continued

16. Chi-Squared Distribution – Example c 2 X 15 χ′

17. f(x) f(x) f(x) x x x Example Solution P(7.261 < X < 24.996) = 0.95-0.05 =0.9 (b)P(X<6.262)= 0.025 χ′ For which P(X < χ′) =0.02 χ′ = 5.985 24.996 7.261 6.262 P χ′

18. F-Distribution

19. F-Distribution – Probability Density Function 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 <  0 , elsewhere Note : 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.

20. 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.

21. 6 and 24 d.f. 6 and 10 d.f. x 0 F-Distribution – Probability Density Function Shapes Probability density functions for various values of ν1 and ν2 f(x)

22. F-Distribution (p=0.01) Table

23. F-Distribution (p=0.05) Table γ1

24. F-Distribution – Table of Probabilities The fp is the f value above which we find an area equal to p, illustrated by the shaded area below. 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. f(x) p x

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

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

27. Example Solution P(Y < 3.09) = F(3.09) = 1- P(Y > 3.09) = 1 - 0.05 =0.95 (b) P(Y > y′ ) = 0.01 y′ =5.07 f(y) y p 3.09 f(y) y 0.01 y′