Outline Continuous Probability Distributions The Uniform - PDF document

12/14/2006 219323 Probability y and Statistics for Software and Knowledge Engineers Lecture 5: Continuous Probability Distributions Monchai Sopitkamon, Ph.D. Outline � Continuous Probability Distributions – The Uniform Distribution (4.1) ( ) – The Exponential Distribution (4.2) – The Gamma Distribution (4.3) – The Weibull Distribution (4.4) – The Beta Distribution (4.5) � Normal Distribution � Normal Distribution 1

12/14/2006 Continuous Probability Distributions: The Uniform Distribution I (4.1) � A RV X w/ a flat pdf between two A RV X w/ a flat pdf between two points, a and b , so that 1 = ( ) f x − b a for a ≤ x ≤ b and f ( x ) = 0 elsewhere � X can be written as X ∼ U ( a , b ) b itt U ( b ) X X − x a � The cdf is for a ≤ x ≤ b = F ( x ) − b a − + 2 ( b a ) a b = = Var ( X ) ( ) E X 12 2 Continuous Probability Distributions: The Uniform Distribution II (4.1) Figure 4 .1 Probability density function of a U( a, b) distribution 2

12/14/2006 Continuous Probability Distributions: The Uniform Distribution III (4.1) − 2 ( 10 0 ) � Ex.30 pg.186 = = Var ( X ) 8 . 33 12 SD ( σ ) = √ 8.33 = 2.89 mm … (cont.) SD ( σ ) = √ 8 33 = 2 89 mm (cont ) Outline � Continuous Probability Distributions – The Uniform Distribution (4.1) ( ) – The Exponential Distribution (4.2) – The Gamma Distribution (4.3) – The Weibull Distribution (4.4) – The Beta Distribution (4.5) � Normal Distribution � Normal Distribution 3

12/14/2006 Continuous Probability Distributions: The Exponential Distribution I (4.2) � Used to model failure or waiting times Used to model failure or waiting times and interarrival times � The pdf is for x ≥ 0, λ > 0 − λ = λ x f ( x ) e and f ( x ) = 0 for x < 0 = 1 − − λ � The cdf is for x ≥ 0 x F ( x ) e 1 1 = = and E ( X ) Var ( X ) λ λ 2 Continuous Probability Distributions: The Exponential Distribution II (4.2) Figure 4 .3 Figure 4 .4 pdf of an exponential p p pdf of an exponential p p distribution w ith λ = 1 distribution w ith λ = 1 / 2 4

12/14/2006 Continuous Probability Distributions: The Memoryless Property of the Exponential Distribution I (4.2.2) � Given that P ( X ≥ x ) = 1 – F ( x ) = e - λ x Then if the RV Y represents the Then, if the RV Y represents the additional time beyond x 0 that elapses before the event occurs, P ( Y ≥ y ) = P ( X ≥ x 0 + y | X ≥ x 0 ) ≥ + − λ + ( x y ) ( ) P X x y e 0 = = = = = = − λ y y 0 e e − λ ≥ x P ( X x 0 ) e 0 so that Y also has an exponential distribution with parameter λ Continuous Probability Distributions: The Memoryless Property of the Exponential Distribution II (4.2.2) Figure 4 .5 I llustration of the m em oryless property of the exponential property of the exponential distribution. The part of the probability density function beyond x 0 is a scaled version of the w hole probability density function 5

12/14/2006 Continuous Probability Distributions: The Memoryless Property Implications (4.2.2) � Exponential distribution is the only continuous distribution with the memoryless property t � Thus exponential distribution is suitable for modeling failure or waiting time where time elapses have no effect on the prob, e.g., to model the failure time of an electronic part that fails due to voltage fluctuation. � However, if the failure is due to wear out, then the memoryless property is not then the memoryless property is not plausible and exponential distribution should not be used to model the failure times. Continuous Probability Distributions: The Poisson Process I (4.2.3) � A Poisson process w/ parameter λ is a process where the distributions of p the time intervals between the occurrences of adjacent events are independent RVs having exponential distributions w/ parameters λ 6

12/14/2006 Continuous Probability Distributions: The Poisson Process II (4.2.3) � Can be used to model – the arrival of calls at a switchboard th i l f ll t it hb d – the addition of new elements to a queue – the arrival of network packets to a switch � The expected waiting time between � The expected waiting time between two events in a Poisson process = the expected value of an exponential distribution w/ parameter λ = 1/ λ Continuous Probability Distributions: The Poisson Process III (4.2.3) � If RV X counts the number of events If RV X counts the number of events occurring within a fixed time interval of length t , then X ∼ P ( λ t .) t λ − λ x e ( t ) = = P ( X x ) x ! 7

12/14/2006 Continuous Probability Distributions: Examples of the Exponential Distribution (4.2.4) � Ex.31, pg.192 Figure 4 .8 Probability density function for shipw reck hunt (see Excel Spreadsheet) Outline � Continuous Probability Distributions – The Uniform Distribution (4.1) ( ) – The Exponential Distribution (4.2) – The Gamma Distribution (4.3) – The Weibull Distribution (4.4) – The Beta Distribution (4.5) � Normal Distribution � Normal Distribution 8

12/14/2006 Continuous Probability Distributions: The Gamma Distribution I (4.3) � Used in the area of reliability theory and analysis of Poisson process � The Gamma Function � The Gamma Function ∞ ∫ Γ = − − k 1 x ( k ) x e dx 0 for k > 0 Note that Γ (1) = 1 and Γ (1/2) = √π , and ( ) ( ) i in general, l Γ ( k ) = ( k – 1) Γ ( k – 1) for k > 1 If n is a positive integer, then Γ ( n ) = ( n – 1)! Continuous Probability Distributions: The Gamma Distribution II (4.3) � The Gamma Distribution w/ The Gamma Distribution w/ parameter k > 0 and λ > 0 has a λ − − λ pdf k k 1 x x e k : shape parameter = f ( x ) λ : scale parameter Γ ( k ) for x ≥ 0 and f ( x ) = 0 for x < 0 k = � Expectation E ( X ) λ k = � Variance Var ( X ) λ 2 9

12/14/2006 Continuous Probability Distributions: The Gamma Distribution III (4.3) Figure 4 .1 3 Probability density functions of gam m a distributions Continuous Probability Distributions: The Gamma Distribution IV (4.3) � If X 1 , …, X k are independent RVs each having an exponential dist w/ each having an exponential dist w/ parameter λ , then the RV X = X 1 + …+ X k has a gamma dist w/ parameters k and λ � This implies that for a Poisson Thi i li th t f P i process w/ parameter λ , the time taken for k events to occur has a gamma dist w/ parameters k and λ 10

12/14/2006 Continuous Probability Distributions: The Gamma Distribution V (4.3) k 5 = λ = = � Ex.32 pg.200: E ( X ) 1 . 16 m λ 4 4 . 3 3 Figure 4 .1 5 Distance to fifth fracture has a gam m a distribution w ith param eters k = 5 and λ = 4 . 3 (see Excel spreadsheet) Outline � Continuous Probability Distributions – The Uniform Distribution (4.1) ( ) – The Exponential Distribution (4.2) – The Gamma Distribution (4.3) – The Weibull Distribution (4.4) – The Beta Distribution (4.5) � Normal Distribution � Normal Distribution 11

12/14/2006 Continuous Probability Distributions: The Weibull Distribution I (4.4) � Used to model failure and waiting times shape parameter scale parameter − − λ = λ a � The pdf is a a 1 ( x ) f ( x ) a x e with parameters a > 0 and λ > 0, for x ≥ 0 and f ( x ) = 0 for x < 0 = − − λ a � The cdf is ( ) x F ( x ) 1 e f for x ≥ 0 0 ⎛ + ⎞ 1 1 � Expectation = Γ ⎜ ⎟ E ( X ) 1 λ ⎝ ⎠ a ⎧ ⎫ 2 ⎪ ⎛ + ⎞ ⎛ + ⎞ ⎪ 1 2 1 � Variance = Γ − Γ ⎜ ⎟ ⎜ ⎟ ⎨ ⎬ Var ( X ) 1 1 λ ⎪ ⎝ ⎠ ⎝ ⎠ ⎪ 2 a a ⎩ ⎭ Continuous Probability Distributions: The Weibull Distribution II (4.4) Figure 4.17 Probability density functions of the Weibull distribution 12

12/14/2006 Continuous Probability Distributions: The Weibull Distribution III (4.4) � Ex.33 pg.204: Figure 4 1 9 Figure 4 .1 9 Distribution of bacteria survival tim es (see Excel spreadsheet) Outline � Continuous Probability Distributions – The Uniform Distribution (4.1) ( ) – The Exponential Distribution (4.2) – The Gamma Distribution (4.3) – The Weibull Distribution (4.4) – The Beta Distribution (4.5) � Normal Distribution � Normal Distribution 13

12/14/2006 Continuous Probability Distributions: The Beta Distribution I (4.5) � Used to model proportions Γ + � The pdf ( a b ) = − − − a 1 b 1 f ( x ) x ( 1 x ) Γ Γ ( a ) ( b ) when a > 0 and b > 0 are parameters for 0 ≤ x ≤ 1, and f ( x ) = 0 elsewhere a a � Expectation E i = E ( X ) + a b ab � Variance = Var ( X ) + + + 2 ( a b ) ( a b 1 ) Continuous Probability Distributions: The Beta Distribution II (4.5) Figure 4 .2 2 Probability density functions of the beta distribution 14

12/14/2006 Continuous Probability Distributions: The Beta Distribution II (4.5) 5 . 5 � Ex. 35 pg.207: = = E ( X ) 0 . 57 + + 5 5 . . 5 5 4 4 . . 2 2 (see Excel spreadsheet) 15