Other Continuous Probability Distributions

Size: px

Start display at page:

Download "Other Continuous Probability Distributions"

Shonda Holmes
5 years ago
Views:

CHAPTER Probability, Statistics, and Reliability for Engineers and Scientists Second Edition PROBABILITY DISTRIBUTION FOR CONTINUOUS RANDOM VARIABLES A. J.

1 CHAPTER Probability, Statistics, and Reliability for Engineers and Scientists Second Edition PROBABILITY DISTRIBUTION FOR CONTINUOUS RANDOM VARIABLES A. J. Clar School of Engineering Department of Civil and Environmental Engineering 5c Probability and Statistics for Civil Engineers Department of Civil and Environmental Engineering University of Maryland, College Par CHAPMAN HALL/CRC CHAPTER 5c. PROBABILITY DISTRIBUTION FOR CONTINUOUS RAND. VARIABLES Slide No. These distribution are classified as Distribution used in statistical analyses Student-t, or t Distribution F Distribution Chi-square (χ ) Distribution Extreme Value Distribution Type I Type II Type III

2 CHAPTER 5c. PROBABILITY DISTRIBUTION FOR CONTINUOUS RAND. VARIABLES Slide No. Student-t, or t Distribution The student-t or t distribution is a symmetric, bell-shaped distribution with the following density function () [( + ) ] f t < t T where ( ) ( )[ ( )] ( + π / + t / ) is a parameter,and the gamma function (). ( n) 0 x e n x dx for any valuen For >, the mean and variance are given, respectively, by T 0 and σt is < + CHAPTER 5c. PROBABILITY DISTRIBUTION FOR CONTINUOUS RAND. VARIABLES Slide No. 3 The gamma function has the following useful properties: ( n) ( n ) ( n ) ( n) ( n )! for integer n As increases toward infinity, the variance of the t distribution approaches unity, and therefore it approaches the standard normal distribution

3 CHAPTER 5c. PROBABILITY DISTRIBUTION FOR CONTINUOUS RAND. VARIABLES Slide No. 4 Properties of the t distribution It is of interest in statistical analysis to determine the percentage points t α, that correspond to the following probability: α P ( T > t ) or α f () t α, T tα, t The percentage points are provided in Table A- of the Textboo. For the lower tail, the following relationship can be used: t α, tα, dt CHAPTER 5c. PROBABILITY DISTRIBUTION FOR CONTINUOUS RAND. VARIABLES Slide No. 5 Critical Values for t Distribution Degrees of Freedom, K Level of Significance,α f T () t 0 t α, ( T t ) area α P > α, t 3

4 CHAPTER 5c. PROBABILITY DISTRIBUTION FOR CONTINUOUS RAND. VARIABLES Slide No. 6 Example: Student-t Distribution. Find P(-t 0.05,0 < T < t 0.05,0 ). Find t such that P(t < T < -.76) 0.045, and 4 Since t 0.05,0 leaves an area of 0.05 to the right and t 0.05,0 leaves an area Of 0.05 to the left, therefore, ( t < T < t ) P 0.05,0 0.05, 0 From the table,.76 corresponds to t 0.05, 4 when 4 Therefore, -t 0.05, Since t in the original probability statement Is to the left of t 0.05, , let t -t α,4. Then from the following figure we have α CHAPTER 5c. PROBABILITY DISTRIBUTION FOR CONTINUOUS RAND. VARIABLES Slide No. 7 Example: Student-t Distribution Or α f T () t area From the table with 4 t t.977 Thus, P t t (.977 < T <.76) t α, t 4

5 CHAPTER 5c. PROBABILITY DISTRIBUTION FOR CONTINUOUS RAND. VARIABLES Slide No. 8 The F Distribution The F distribution has two shape parameters ν and ν u, and has the following PDF: u + ( f ) ( ) u f F f for f u+ u f + u The mean and variance are given by F u u and σ F u ( u + ) ( u ) ( u 4) > 0 for u > 4 CHAPTER 5c. PROBABILITY DISTRIBUTION FOR CONTINUOUS RAND. VARIABLES Slide No. 9 The F Distribution The distribution is positively sewed with a shape that depends on and u. It is of interest in statistical analysis to determine the percentage points f α,,u that correspond to the following probability: P( F > f ) f ( x) dx α α, u α, F f α,, u 5

6 CHAPTER 5c. PROBABILITY DISTRIBUTION FOR CONTINUOUS RAND. VARIABLES Slide No. 0 The F Distribution The percentage points are provided in Table A- 4 of the Textboo. The F distribution has a unique property that allows tabulating values for the upper tail only. For the lower tail, the following relation can be used to find the percentage points: f α,, u f Note : f α, u, α,, u f α,, u CHAPTER 5c. PROBABILITY DISTRIBUTION FOR CONTINUOUS RAND. VARIABLES Slide No. The F Distribution Upper values for 5% (First row) and % (second row) Significance Level α Second degrees of freedom, u First degrees of freedom,

7 CHAPTER 5c. PROBABILITY DISTRIBUTION FOR CONTINUOUS RAND. VARIABLES Slide No. Chi-square (χ ) Distribution This distribution is frequently encountered in statistical analysis, where we deal with the sum of squares of random variables with standard normal distribution, χ C Z + Z Z Where C random variable with chisquare, and Z to Z are normally distributed (standard normal) CHAPTER 5c. PROBABILITY DISTRIBUTION FOR CONTINUOUS RAND. VARIABLES Slide No. 3 Chi-square (χ ) Distribution The probability density function (PDF) of the chi-square distribution is 0.5 fc () c c e for c > The mean and variance are given, respectively, by C and σ C c 7

8 CHAPTER 5c. PROBABILITY DISTRIBUTION FOR CONTINUOUS RAND. VARIABLES Slide No. 4 Chi-square (χ ) Distribution This distribution is positively sewed with a shape that depends on the parameter. It is of interest in statistical analysis to determine the percentage points c α, that correspond to the following probability: α P( C > c ) f () c dc α, cα, These percentage points are provided in Table A-3 of the Textboo. C CHAPTER 5c. PROBABILITY DISTRIBUTION FOR CONTINUOUS RAND. VARIABLES Slide No. 5 Extreme Value In many engineering applications, the extreme values of random variables are of special importance. The largest or smallest values of random variables may dictate a particular design. Wind speeds, for example, are recorded continuously at airports and weather stations. The maximum wind speeds per hour, month, day, year, or other period can be used for this purpose 8

9 CHAPTER 5c. PROBABILITY DISTRIBUTION FOR CONTINUOUS RAND. VARIABLES Slide No. 6 Extreme Value Usually, the information on yearly maximum wind speed is used in engineering profession. If the design wind speed has a 50-year return period, then the probability that the wind speed will exceed the design value in a year is / Design of earthquae loads, flood levels, and so forth are also determined in this manner. CHAPTER 5c. PROBABILITY DISTRIBUTION FOR CONTINUOUS RAND. VARIABLES Slide No. 7 Extreme Value In some cases, the minimum value of a random variable is also of interest for design applications. For example, when a large number of identical devices, such as calculators or cars, are manufactured, their minimum service lives are of great interest to consumers. In constructing an extreme value distribution, an underlying random variable with a particular distribution is necessary. 9

10 CHAPTER 5c. PROBABILITY DISTRIBUTION FOR CONTINUOUS RAND. VARIABLES Slide No. 8 Extreme Value If different sets of samples are obtained (through physical or numerical experimentation), one can select the extreme values from each sample set, either the maximum or the minimum values, and then construct a different distribution for the extreme values. Therefore, the underlying distribution of a variable governs the form of the corresponding extreme value distribution. CHAPTER 5c. PROBABILITY DISTRIBUTION FOR CONTINUOUS RAND. VARIABLES Slide No. 9 Engineering Significance of Extreme Values In structural reliability and safety, the maximum loads and low structural resistance are the values most relevant to assure safety or reliability of a structure. The prediction of future conditions is often required in engineering design, and may involve the prediction of the largest or smallest value. 0

11 CHAPTER 5c. PROBABILITY DISTRIBUTION FOR CONTINUOUS RAND. VARIABLES Slide No. 0 Engineering Significance of Extreme Values Therefore, extrapolation from previously observed extreme value data is invariably necessary. The asymptotic theory provides a powerful basis for developing the required engineering information. CHAPTER 5c. PROBABILITY DISTRIBUTION FOR CONTINUOUS RAND. VARIABLES Slide No. Type I Extreme Value Two forms of the Type I extreme value distribution can be used: The largest extreme value The smallest extreme value

12 CHAPTER 5c. PROBABILITY DISTRIBUTION FOR CONTINUOUS RAND. VARIABLES Slide No. The Type I Extreme Value Distribution (Largest) The probability density function (PDF) of the f X n Type I for largest distribution is αn x un αn x un x α e exp e [ ] ( ) ( ) ( ) The CDF is given by F X n [ ] αn ( x un ) ( x) exp e The mean and variance are given, respectively, by X n u n + n γ α n and σ X n π 6α n (γ ) CHAPTER 5c. PROBABILITY DISTRIBUTION FOR CONTINUOUS RAND. VARIABLES Slide No. 3 The Type I Extreme Value Distribution (Smallest) The probability density function (PDF) of the Type I for smallest distribution is α x u α x u f x α e exp e X [ ] ( ) ( ) ( ) The CDF is given by F X [ ] α ( x u ) ( x) exp e The mean and variance are given, respectively, by X u γ α and σ X π 6α (γ )

13 CHAPTER 5c. PROBABILITY DISTRIBUTION FOR CONTINUOUS RAND. VARIABLES Slide No. 4 Applications of Type I Distribution Strength of brittle materials (Johnson 953) can be described by Type I smallest value Hydrological phenomena such as the maximum daily flow in a year or the annual pea flow hourly discharge during flood (Chow 95) Wind maximum velocity in a year. CHAPTER 5c. PROBABILITY DISTRIBUTION FOR CONTINUOUS RAND. VARIABLES Slide No. 5 Example: Type I Largest The data on maximum wind velocity V n at a site have been compiled for n years, and its mean and standard deviation are estimated to be 6.3 mph and 7.5 mph, respectively. Assuming that V n has a Type I extreme value distribution, what is the probability that the maximum wind velocity will exceed 00 mph in any given year? 3

14 CHAPTER 5c. PROBABILITY DISTRIBUTION FOR CONTINUOUS RAND. VARIABLES Slide No. 6 Example (cont d): Type I Largest π 6σ α n X n 6( 7.5) The parameters u n and α n can be calculated as π and u n X n The probability that the maximum wind velocity is greater than 00 mph is P αn ( x un ) ( X > 00) F ( x) exp e n X exp γ α n n [ ] ( ) [ ] e CHAPTER 5c. PROBABILITY DISTRIBUTION FOR CONTINUOUS RAND. VARIABLES Slide No. 7 Example: Type I Largest Suppose that in the previous example the design wind speed with a return period of 00 years needs to be estimated for a particular site. With V d denoted as the design wind speed to be estimated, the probability that it will be exceeded in a given year is / Thus, P ( X > V ) F ( V ) or n - exp - d X n d ( Vd ) [ e ] 0.0 Vd mph 4

15 CHAPTER 5c. PROBABILITY DISTRIBUTION FOR CONTINUOUS RAND. VARIABLES Slide No. 8 The Type II Extreme Value Distribution Largest Smallest The Type III Extreme Value Distribution Largest Smallest CHAPTER 5c. PROBABILITY DISTRIBUTION FOR CONTINUOUS RAND. VARIABLES Slide No. 9 Type II Largest and F f Y Y ( y) ( y) e v v y v y + e for y 0 v y 5

16 CHAPTER 5c. PROBABILITY DISTRIBUTION FOR CONTINUOUS RAND. VARIABLES Slide No. 30 Type II Largest Y v for > σ Y v COV for > CHAPTER 5c. PROBABILITY DISTRIBUTION FOR CONTINUOUS RAND. VARIABLES Slide No. 3 Gamma Function Properties 0 t () t r exp( r) ( t) ( t ) ( t ) For an integer n, the gamma function becomes the factorial: ( n) ( n )! dr Appendix B of the Textboo contains tabulated values of the gamma function 6

17 CHAPTER 5c. PROBABILITY DISTRIBUTION FOR CONTINUOUS RAND. VARIABLES Slide No. 3 Extreme Value Distribution Coefficient of Variation versus the parameter (Benjamin and Cornell, 970) CHAPTER 5c. PROBABILITY DISTRIBUTION FOR CONTINUOUS RAND. VARIABLES Slide No. 33 Example, Type II: Wind Velocity In Boston, Massachusetts, the measured data suggest that the mean and standard deviation of the maximum annual wind velocity are 55 mph and.8 mph, respectively. What is the velocity y which will be exceeded with a probability value of 0.0? 7

18 CHAPTER 5c. PROBABILITY DISTRIBUTION FOR CONTINUOUS RAND. VARIABLES Slide No. 34 Example, Type II (cont d): Wind Velocity.8 COV From Fig., 6.5 Y v ( x + ) x( x) ( x) Y 55 v 6.5 ( x + ) x (.846) ( 0.846) ( ) CHAPTER 5c. PROBABILITY DISTRIBUTION FOR CONTINUOUS RAND. VARIABLES Slide No. 35 Example, Type II (cont d): Wind Velocity Y - F e or v Y ( y) 49.4 y y 9mph Y 55 v mph 8

19 9 CHAPTER 5c. PROBABILITY DISTRIBUTION FOR CONTINUOUS RAND. VARIABLES Slide No. 36 Type II Smallest ( ) ( ) z v Z z v Z e z v v z f z e z F + and 0 for CHAPTER 5c. PROBABILITY DISTRIBUTION FOR CONTINUOUS RAND. VARIABLES Slide No. 37 Type II Smallest for for > > COV v v Z Z σ

20 CHAPTER 5c. PROBABILITY DISTRIBUTION FOR CONTINUOUS RAND. VARIABLES Slide No. 38 Type III Largest Smallest Most useful applications of this model deal with smallest values. CHAPTER 5c. PROBABILITY DISTRIBUTION FOR CONTINUOUS RAND. VARIABLES Slide No. 39 Type III F f Smallest Z Z ( z) z ω exp for z ω u ω z ω z ω u ω u ω u ω ( z) exp for z ω 0

21 CHAPTER 5c. PROBABILITY DISTRIBUTION FOR CONTINUOUS RAND. VARIABLES Slide No. 40 Type III Smallest ( ) ( ) u u Z Z ω σ ω ω

Theoretical Probability Models

CHAPTER Duxbury Thomson Learning Maing Hard Decision Third Edition Theoretical Probability Models A. J. Clar School of Engineering Department of Civil and Environmental Engineering 9 FALL 003 By Dr. Ibrahim.