Theoretical Probability Models
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1 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. Assaaf ENCE 67 Decision Analysis for Engineering Department of Civil and Environmental Engineering University of Maryland, College Par CHAPTER 9. THEORETICAL PROBABILITY MODELS 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 9. THEORETICAL PROBABILITY MODELS Slide No. Student-t, or t Distribution The student-t or t distribution is a symmetric, bell-shaped distribution with the following density function () [( + ) ] ft t < t π where ( ) ( / ) [ + ( t / ) ] ( ) is a parameter, and the gamma function (). ( n) 0 x e n x dx for any value n For >, the mean and variance are given, respectively, by T 0 and σt is < + CHAPTER 9. THEORETICAL PROBABILITY MODELS 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 9. THEORETICAL PROBABILITY MODELS 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 dt The percentage points are usually provided in tables. For the lower tail, the following relationship can be used: t t α, α, CHAPTER 9. THEORETICAL PROBABILITY MODELS 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
4 CHAPTER 9. THEORETICAL PROBABILITY MODELS 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 9. THEORETICAL PROBABILITY MODELS Slide No. 7 Example: Student-t Distribution Or α area f T ( t) From the table with 4 t Thus, P t t t (.977 < T <.76) t α, t
5 CHAPTER 9. THEORETICAL PROBABILITY MODELS 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 9. THEORETICAL PROBABILITY MODELS 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
6 CHAPTER 9. THEORETICAL PROBABILITY MODELS Slide No. 30 The F Distribution The percentage points are usually provided in mathematical boo tables. 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 9. THEORETICAL PROBABILITY MODELS Slide No. 3 The F Distribution Second degrees of freedom, u First degrees of freedom, Upper values for 5% (First row) and % (second row) Significance Level α
7 CHAPTER 9. THEORETICAL PROBABILITY MODELS Slide No. 3 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 9. THEORETICAL PROBABILITY MODELS Slide No. 33 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
8 CHAPTER 9. THEORETICAL PROBABILITY MODELS Slide No. 34 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 α, cα, These percentage points are usually provided in tables. C dc CHAPTER 9. THEORETICAL PROBABILITY MODELS Slide No. 35 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
9 CHAPTER 9. THEORETICAL PROBABILITY MODELS Slide No. 36 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 9. THEORETICAL PROBABILITY MODELS Slide No. 37 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.
10 CHAPTER 9. THEORETICAL PROBABILITY MODELS Slide No. 38 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. CHAPTER 9. THEORETICAL PROBABILITY MODELS Slide No. 39 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.
11 CHAPTER 9. THEORETICAL PROBABILITY MODELS Slide No. 40 Type I Extreme Value Two forms of the Type I extreme value distribution can be used: The largest extreme value The smallest extreme value CHAPTER 9. THEORETICAL PROBABILITY MODELS Slide No. 4 The Type I Extreme Value Distribution (Largest) The probability density function (PDF) of the f X n Type I for largest distribution is αn xun αn xun x α e exp e [ ] ( ) ( ) ( ) The CDF is given by F X n [ ] αn ( xun ) ( 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 )
12 CHAPTER 9. THEORETICAL PROBABILITY MODELS Slide No. 4 The Type I Extreme Value Distribution (Smallest) The probability density function (PDF) of the Type I for smallest distribution is α xu α xu f x α e exp e X [ ] ( ) ( ) ( ) The CDF is given by F X [ ] α ( xu ) ( x) exp e The mean and variance are given, respectively, by X u γ α and σ X π 6α (γ ) CHAPTER 9. THEORETICAL PROBABILITY MODELS Slide No. 43 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.
13 CHAPTER 9. THEORETICAL PROBABILITY MODELS Slide No. 44 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? CHAPTER 9. THEORETICAL PROBABILITY MODELS Slide No. 45 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 αn xun P X > 00 F x exp e ( ) ( ) ( ) n X exp γ α n n [ ] ( ) [ ] e
14 CHAPTER 9. THEORETICAL PROBABILITY MODELS Slide No. 46 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 CHAPTER 9. THEORETICAL PROBABILITY MODELS Slide No. 47 The Type II Extreme Value Distribution Largest Smallest The Type III Extreme Value Distribution Largest Smallest
15 CHAPTER 9. THEORETICAL PROBABILITY MODELS Slide No. 48 Type II Largest ( ) ( ) y v Y y v Y e y v v y f y e y F + and 0 for CHAPTER 9. THEORETICAL PROBABILITY MODELS Slide No. 49 Type II Largest for for > > COV v v Y Y σ
16 CHAPTER 9. THEORETICAL PROBABILITY MODELS Slide No. 50 Gamma Function Properties 0 t () t r exp( r) ( t) ( t ) ( t ) ( n) ( n )! Mathematical handboos usually contain tabulated values of the gamma function For an integer n, the gamma function becomes the factorial: dr CHAPTER 9. THEORETICAL PROBABILITY MODELS Slide No. 5 Extreme Value Distribution Coefficient of Variation versus the parameter (Benjamin and Cornell, 970)
17 CHAPTER 9. THEORETICAL PROBABILITY MODELS Slide No. 5 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? CHAPTER 9. THEORETICAL PROBABILITY MODELS Slide No. 53 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) ( )
18 CHAPTER 9. THEORETICAL PROBABILITY MODELS Slide No. 54 Example, Type II (cont d): Wind Velocity Y v - F e or Y ( y) 49.4 y y 9mph Y 55 v mph CHAPTER 9. THEORETICAL PROBABILITY MODELS Slide No. 55 Type II Smallest and F f Z Z ( z) ( z) e v v z v z + e for z 0 v z
19 CHAPTER 9. THEORETICAL PROBABILITY MODELS Slide No. 56 Type II Smallest Z v for > σ Z v COV for > CHAPTER 9. THEORETICAL PROBABILITY MODELS Slide No. 57 Type III Largest Smallest Most useful applications of this model deal with smallest values.
20 CHAPTER 9. THEORETICAL PROBABILITY MODELS Slide No. 58 Type III Smallest ( ) ( ) z u z u z u z f z u z z F Z Z for exp for exp CHAPTER 9. THEORETICAL PROBABILITY MODELS Slide No. 59 Type III Smallest ( ) ( ) u u Z Z σ
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