Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Chapter 4-1 Continuous Random Variables 主講人:虞台文 Content Random Variables and Distribution Functions Probability Density Functions of Continuous Random Variables The Exponential Distributions The Reliability and Failure Rate The Erlang Distributions The Gamma Distributions The Gaussian or Normal Distributions The Uniform Distributions Chapter 4-1 Continuous Random Variables Random Variables and Distribution Functions The Temperature in Taipei 今天中午台北市氣溫為 25C之機率為何? 今天中午台北市氣溫小於 或等於25C之機率為何? Renewed Definition of Random Variables A random variable X on a probability space (, A, P) is a function X : R that assigns a real number X() to each sample point , such that for every real number x, the set {|X() x} is an event, i.e., a member of A. The (Cumulative) Distribution Functions The (cumulative) distribution function FX of a random variable X is defined to be the function FX(x) = P(X x), − < x < . Example 1 Example 1 x p X ( x) 0 1 2 3 1 8 3 8 3 8 1 8 F X (x ) 1 0 1 8 FX ( x) 84 7 8 1 x0 0 x 1 1 x 2 2 x3 3 x x -3 -2 -1 0 1 2 3 4 5 6 7 Example 1 FY ( y) P(Y y) y R2 2 R y 2 y 2 R 0 yR Example 1 FY ( y) P(Y y) 0 2 y 2 R 1 y0 0 yR 1 y R y Example 1 FY ( y) P(Y y) 0 2 y 2 R 1 y0 F Y (y ) 1 0 yR 0.5 1 y y 0 R /2 R Example 1 FZ ( z ) P( Z z ) 0 ( R z )2 1 R2 5 98 9 1 ( R z )2 R2 1 RY z0 R 0 z R 3 R 3 z R 2 R 2 z 2R 3 2R 3 zR Rz R/2 Example 1 FZ ( z ) P( Z z ) 0 ( R z )2 1 R2 5 98 9 1 ( R z )2 R2 1 z0 F Z (y ) 1 0 z R 3 R 3 z R 2 R 2 z 2R 3 2R 3 zR Rz 0.5 z 0 R /3 R /2 2R /3 R Example 1 FX ( x) 11 FY ( y ) FFXX(x(x)) FZ ( z ) FFZ (y(y) ) Z FFY (y(y) ) Y 11 11 0.5 0.5 0.5 0.5 xx -3 -3 -2 -2 -1 -1 00 11 22 33 44 55 66 77 00 yy RR/2/2 RR 00 zz R /3 R /2 2R /3 R /3 R /2 2R /3 R R Properties of Distribution Functions 0 F(x) 1 for all x; F is monotonically nondecreasing; F() = 0 and F() =1; F(x+) = F(x) for all x. 1. 2. 3. 4. 11 FFXX(x(x)) FFZ (y(y) ) Z FFY (y(y) ) Y 11 11 0.5 0.5 0.5 0.5 xx -3 -3 -2 -2 -1 -1 00 11 22 33 44 55 66 77 00 yy RR/2/2 RR 00 zz R /3 R /2 2R /3 R /3 R /2 2R /3 R R Definition Continuous Random Variables A random variable X is called a continuous random variable if P ( X x ) F ( x ) F ( x ) 0 11 FFXX(x(x)) FFZ (y(y) ) Z FFY (y(y) ) Y 11 11 0.5 0.5 0.5 0.5 xx -3 -3 -2 -2 -1 -1 00 11 22 33 44 55 66 77 00 yy RR/2/2 RR 00 zz R /3 R /2 2R /3 R /3 R /2 2R /3 R R Example 2 11 FFXX(x(x)) FFZ (y(y) ) Z FFY (y(y) ) Y 11 11 0.5 0.5 0.5 0.5 xx -3 -3 -2 -2 -1 -1 00 11 22 33 44 55 66 77 00 yy RR/2/2 RR 00 zz R /3 R /2 2R /3 R /3 R /2 2R /3 R R Chapter 4-1 Continuous Random Variables Probability Density Functions of Continuous Random Variables Probability Density Functions of Continuous Random Variables A probability density function (pdf) fX(x) of a continuous random variable X is a nonnegative function f such that x FX ( x) f X (u)du Probability Density Functions of Continuous Random Variables A probability density function (pdf) fX(x) of a continuous random variable X is a nonnegative function f such that x FX ( x) f X (u)du x FX ( x) f X (u)du Properties of Pdf's 1. f ( x) 0; Remark: f(x) can be larger than 1. 2. f ( x)dx 1; 3. P(a X b) P(a X b) P(a X b) P(a X b) P( X b) P( X a) F (b) F (a) b a b a f ( x)dx f ( x)dx f ( x)dx dF ( x) 4. f ( x) F ( x). dx Example 3 k 6 Example 3 kx(1 x) 0 x 1 f ( x) otherwise 0 k 6 1 x x k 1 f ( x)dx kx(1 x)dx k 0 2 3 0 6 1 2 3 k 6 Example 3 6 x(1 x) 0 x 1 f ( x) otherwise 0 x0 0 x0 0 x 2 3 F ( x) 6u (1 u )du 0 x 1 3x 2 x 0 x 1 0 1 1 x 1 x 1 k 6 Example 3 6 x(1 x) 0 x 1 f ( x) otherwise 0 2 f (x ) 1.5 1 x0 0 F ( x) 3x 2 2 x3 0 x 1 1 1 x 0.5 x 0 -1 0 1.2 1 2 F (x ) 1 0.8 0.6 0.4 0.2 x 0 -1 0 1 2 k 6 Example 3 6 x(1 x) 0 x 1 f ( x) otherwise 0 2 f (x ) 1.5 1 x0 0 F ( x) 3x 2 2 x3 0 x 1 1 1 x 0.5 x 0 -1 0 1.2 1 2 F (x ) 1 0.8 P( X 13 ) 3( 13 ) 2 2( 13 )3 7 0.25926 27 0.6 0.4 0.25926 0.2 x 0 -1 0 1/3 1 2 Chapter 4-1 Continuous Random Variables The Exponential Distributions The Exponential Distributions The following r.v.’s are often modelled as exponential: 1. Interarrival time between two successive job arrivals. 2. Service time at a server in a queuing network. 3. Life time of a component. The Exponential Distributions A r.v. X is said to possess an exponential distribution and to be exponentially distributed, denoted by X ~ Exp(), if it possesses the density f ( x) e x , x0 : arriving rate : failure rate The Exponential Distributions X ~ Exp( ) pdf f ( x) e x 1 e cdf F ( x) 0 , x0 x x0 x0 3.5 : arriving rate : failure rate f (x ) 3 2.5 2 X ~ Exp (2) 1.5 1 The Exponential Distributions 0.5 x 0 -2 0 1.2 2 4 6 F (x ) 1 X ~ Exp( ) 0.8 0.6 0.4 0.2 x 0 -2 0 2 4 -0.2 pdf f ( x) e x 1 e cdf F ( x) 0 , x0 x x0 x0 6 X ~ Exp( ) a0 P ( X a b | X a ) P ( X b) b0 Memoryless or Markov Property X ~ Exp( ) a0 P ( X a b | X a ) P ( X b) b0 Memoryless or Markov Property P( X a b | X a ) P( X a b and X a) P( X a ) P ( X a b ) 1 P ( X a b ) 1 F ( a b) P( X a) 1 P( X a ) 1 F (a) e ( a b ) a e b e P ( X b) X ~ Exp( ) pdf f ( x) e x , x 0 1 e x cdf F ( x) 0 x0 x0 X ~ Exp( ) a0 P ( X a b | X a ) P ( X b) b0 Memoryless or Markov Property Exercise: 連續型隨機變數中,唯有指數分佈具備無記憶性。 The Relation Between Poisson and Exponential Distributions : arriving rate : failure rate Let r.v. Nt denote #jobs arriving to a computer system in the interval (0, t]. Nt ~ ?P(t ) Nt 0 t The Relation Between Poisson and Exponential Distributions : arriving rate : failure rate Let r.v. Nt denote #jobs arriving to a computer system in the interval (0, t]. Nt ~ ?P(t ) Nt The next arrival t 0 X Let X denote the time of the next arrival. X ~? f (t ) ? or F (t ) ? P( X t ) P( Nt 0) The Relation Between Poisson and Exponential Distributions : arriving rate : failure rate Let r.v. Nt denote #jobs arriving to a computer system in the interval (0, t]. Nt ~ ?P(t ) Nt 0 能求出P(X > t)嗎? t The next arrival X Let X denote the time of the next arrival. X ~? f (t ) ? or F (t ) ? P( X t ) P( Nt 0) X ~ Exp( ) pdf cdf f ( x) e x , x 0 The Relation Between Poisson and x 1 e x Distributions 0 Exponential : arriving rate F ( x) 0 x0 : failure rate PLet ( X r.v. t )Nt P ( Nt 0) denote #jobs arriving to a computer system t in the interval (t )0(0, e t]. e t t 0 t 0! N N ~ ?P(t ) t t F (t ) 1 e t 0 0 能求出P(X > t)嗎? t 0 f (t ) e t t The next arrival X Let X denote the time of the next arrival. X ~? f (t ) ? or F (t ) ? The Relation Between Poisson and Exponential Distributions : arriving rate : failure rate Let r.v. Nt denote #jobs arriving to a computer system in the interval (0, t]. Nt 0 Nt ~ ?P(t ) The next arrival t X Let X denote the time of the next arrival. X ~ Exp( ) The Relation Between Poisson and Exponential Distributions : arriving rate t1 t2 t3 t4 : failure rate t5 The interarrival times of a Poisson process are exponentially distributed. P(“No job”) = ? 0 Example 5 10 secs = 0.1 job/sec P(“No job”) = ? 0 Example 5 10 secs = 0.1 job/sec Method 1: Let N10 represent #jobs arriving in the 10 secs. P("No job") P( N10 0) 10 e 1 e 1 0! N10 ~ P(1) Method 2: Let X represent the time of the next arriving job. P("No job") P( X 10) 1 P( X 10) 1 (1 e10 ) e 1 X ~ Exp(0.1) Chapter 4-1 Continuous Random Variables The Reliability and Failure Rate Definition Reliability Let r.v. X be the lifetime or time to failure of a component. The probability that the component survives until some time t is called the reliability R(t) of the component, i.e., R(t) = P(X > t) = 1 F(t) Remarks: 1. F(t) is, hence, called unreliability. 2. R’(t) = F’(t) = f(t) is called the failure density function. The Instantaneous Failure Rate The Instantaneous Failure Rate 生命將在時間t後瞬間結束的機率 P(t X t t | X t ) t 0 t t+t F (t t ) F (t ) P(t X t t | X t ) R(t ) The Instantaneous Failure Rate 生命將在時間t後瞬間結束的機率 P(t X t t | X t ) P(t X t t and X t ) P( X t ) P(t X t t ) F (t t ) F (t ) P( X t ) R(t ) F (t t ) F (t ) P(t X t t | X t ) R(t ) The Instantaneous Failure Rate 瞬間暴斃率h(t) P(t X t t | X t ) h(t ) lim t 0 t F (t t ) F (t ) F (t ) lim t 0 tR(t ) R(t ) f (t ) R(t ) The Instantaneous Failure Rate 瞬間暴斃率h(t) f (t ) h(t ) R(t ) X ~ Exp( ) pdf Example 6 f ( x) e x , x 0 1 e x cdf F ( x) 0 x0 x0 Show that the failure rate of exponential distribution is characterized by a constant failure rate. f (t ) e t f (t ) t h(t ) e R(t ) 1 F (t ) t 0 以指數分配來model物件壽命之機率分配合理嗎? More on Failure Rates h(t) CFR h(t ) 0 t More on Failure Rates h(t) IFR DFR h(t ) 0 Useful Life CFR h(t ) 0 h(t ) 0 t More on Failure Rates h(t) DFR h(t ) 0 ? Exponential Distribution Useful Life CFR h(t ) 0 IFR h(t ) 0 ? t Relationships among F(t), f(t), R(t), h(t) 1 F (t ) F (t ) dF (t ) dt f (t ) R (t ) f (t ) R (t ) h(t ) Relationships among F(t), f(t), R(t), h(t) F (t ) f (t ) f (t ) R (t ) 1 F (t ) R (t ) f (t ) h(t ) Relationships among F(t), f(t), R(t), h(t) F (t ) 1 R (t ) d F (t ) dt f (t ) R (t ) f (t ) R (t ) h(t ) Relationships among F(t), f(t), R(t), h(t) F (t ) R (t ) ? f (t ) ? ? h(t ) Cumulative Hazard t H (t ) h( x)dx 0 t 0 t R( x) f ( x) dx dx 0 R( x) R( x) t 1 dR( x) ln R( x) 0 0 R( x) t ln R(t ) ln R(0) ln1 ln R(t ) ln R (t ) R(t ) e H (t ) t 0 e h ( x ) dx Relationships among F(t), f(t), R(t), h(t) F (t ) R (t ) 1 R (t ) f (t ) d F (t ) dt t h ( x ) dx e 0 h(t ) Example 7 2 t x 0t 2 H (t ) h( x)dx 0 xdx 0 , t0 0 0 2 0 2 t 0t 2 R(t ) exp , 2 t t 0 Chapter 4-1 Continuous Random Variables The Erlang Distributions 我的老照相機與閃光燈 它只能使用四次 每使用一次後轉動九十度 使用四次後壽終正寢 The Erlang Distributions time The lifetime of my flash (X) [0, ) I(X)=? fX(t)=? Nt ~ P(t) The Erlang Distributions Consider a component subjected to an environment so that Nt, the number of peak stresses in the interval (0, t], is Poisson distributed with parameter t. Suppose that the rth peak will cause a failure. Let X denote the lifetime of the component. Then, ( t ) k e t P( X t ) ?P( Nt r ) k! k 0 r 1 (t )k e t , cdf F (t ) 1 k! k 0 r 1 t 0 Nt ~ P(t) The Erlang Distributions Exercise of Consider a component subjected to an environment so that Chapter Nt, the number of peak stresses in the interval (0, t],2is Poisson distributed with parameter t. Suppose that the rth peak will cause a failure. Let X denote the lifetime of the component. Then, pdf f (t ) r t r 1et (r 1)! , t 0 (t )k e t , cdf F (t ) 1 k! k 0 r 1 t 0 The r-Stage Erlang Distributions Consider a component subjected to an environment so that Nt, the number of peak stresses in the interval (0, t], is Poisson distributed with parameter t. Suppose that the rth peak will cause a failure. Let X denote the lifetime of the component. Then, pdf f (t ) r t r 1et (r 1)! , t 0 (t )k e t , cdf F (t ) 1 k! k 0 r 1 t 0 Exp( ) Erlang (1, ) The r-Stage Erlang Distributions X Erlang (r , ) pdf f (t ) r t r 1et (r 1)! , t 0 (t )k e t , cdf F (t ) 1 k! k 0 r 1 t 0 Exp( ) Erlang (1, ) The r-Stage Erlang Distributions X Erlang (r , ) t e f (t ) , t 0 r r 1 t pdf r t r 1et (r 1)! (r 1)! , t 0 X Example 8 Erlang (r , ) f (t ) r x r 1e x (r 1)! , x0 In a batch processing environment, the number of jobs arriving for service is 9 per hour. If the arrival process satisfies the requirement of a Poisson experiment. Find the probability that the elapse time between a given arrival and the fifth subsequent arrival is less than 10 minutes. = 9 jobs/hr. Let X represent the time of the 5th arrival. P( X 16 hr) ? 1/ 6 0 5 4 9 x 9 xe 4! X ~ Erlang (5,9) k (9 / 6) 9/ 6 0.0285 dx 1 e k! k 0 4 Chapter 4-1 Continuous Random Variables The Gamma Distributions Review X r為一正整數 欲將之推廣為正實數 Erlang (r , ) pdf f (t ) x e r r 1 x (r 1)! , x0 0 Review X Erlang ( r, ) pdf f (t ) r 1 x r x e , (r 1)!( ) x0 The Gamma Distributions X ( , ), 0 x e pdf f ( x) ( ) 1 x , x0 X ( , ) x e f (t ) ( ) 1 x Review ( ) x 1 x 0 e dx 1. (1) 1 2. ( ) ( 1)( ) 3. ( ) ( 1)!, N 1 4. 2 (n 1)! n 5. n1 n1 2 2 2 ! , x0 X Chi-Square Distributions ( , ) x e f (t ) ( ) 1 x X 1 1 , 2 2 X v 1 , 2 2 2 2 v , x0 Chapter 4-1 Continuous Random Variables The Gaussian or Normal Distributions The Gaussian or Normal Distributions 德國的10馬克紙幣, 以高斯(Gauss, 1777-1855)為 人像, 人像左側有一常態分佈之p.d.f.及其圖形。 The Gaussian or Normal Distributions X N ( , ) 2 1 pdf f ( x) n( x; , ) e 2 2 ( x )2 2 2 x : mean : standard deviation 2: variance The Gaussian or Normal Distributions X N ( , ) Inflection point 2 f ( x) n( x; , ) 2 1 e 2 ( x ) 2 2 x 2 Inflection point : mean : standard deviation 2: variance The Gaussian or Normal Distributions X N ( , ) 2 15 varying f ( x) n( x; , ) 2 1 e 2 ( x )2 2 2 x 100 varying : mean : standard deviation 2: variance The Gaussian or Normal Distributions X N ( , ) 2 Facts: f ( x) n( x; , ) e x2 / 2 dx 2 2 1 e 2 ( x )2 2 x 2 1 2 1 2 e x2 / 2 e dx 1 ( x )2 / 2 2 dx 1 : mean : standard deviation 2: variance The Gaussian or Normal Distributions X N ( , ) e 2 f ( x) n( x; , ) 2 1 e 2 ( x )2 2 x 2 0 e ( x 9)2 /8 x2 /18 1 2 dx 2? 2 3 2 dx ? 2 e ( x )2 / 2 2 dx 1 Standard Normal Distribution X N ( , ) 2 f ( x) n( x; , ) 2 1 e 2 ( x )2 2 2 x Z N (0,1) f Z ( z ) n( z;0,1) 1 z2 / 2 e 2 z Table of N(0, 1) Z N (0,1) f Z ( z ) n( z;0,1) 1 z2 / 2 FZ ( z ) ( z ) e 2 1 z x2 / 2 e dx z 2 z Table of N(0, 1) (1.625) ?0.0521 (1.625) ?0.9479 z Fact: ( z ) 1 ( z ) FZ ( z ) ( z ) 1 z x2 / 2 e dx 2 Probability Evaluation for N(, 2) f ( x) x 1 e 2 ( x )2 x 2 2 1 FX ( x) 2 (t )2 y2 2 2 2 x ( t )2 e 2 2 t 1 dt ? 2 y t y tx t e ( t )2 2 2 dt Probability Evaluation for N(, 2) f ( x) x 1 e 2 ( x )2 2 2 x t y ( t )2 ( t )2 x tx 1 1 2 2 2 2 ? FX ( x) e dt e dt t 2 2 x y x 2 1 1 y2 / 2 y /2 e dy e dy 2 2 y x Fact: X N (0,1) Probability Evaluation for N(, 2) X x N ( , ) 2 x FX ( x) Z-Score:表距離中心若干個標準差 Example 9 X ~ N(12.00, 0.202) 1. P(11.92 X 12.27) ? 2. P( X 12.45) ? 3. P( X 11.70) ? X ~ N(12.00, 0.202) Example 9 P(11.92 X 12.27) P( X 12.27) P( X 11.92) 12.27 12 11.92 12 0.2 0.2 1.35 0.40 0.9115 0.3446 0.5669 1. P(11.92 X 12.27) ?0.5669 2. P( X 12.45) ? 3. P( X 11.70) ? X ~ N(12.00, 0.202) Example 9 P( X 12.45) 1 P( X 12.45) 12.45 12 1 1 2.25 0.2 1 0.9878 0.0122 1. P(11.92 X 12.27) ?0.5669 2. P( X 12.45) ?0.0122 3. P( X 11.70) ? X ~ N(12.00, 0.202) Example 9 11.70 12 1.50 P( X 11.70) 0.2 0.0668 1. P(11.92 X 12.27) ?0.5669 2. P( X 12.45) ?0.0122 3. P( X 11.70) ?0.0668 Example 10 N ( , 2 ) X |X | < |X | < 2 |X | < 3 N ( , 2 ) X Example 10 |X | < |X | < 2 |X | < 3 P(| X | k ) P(k X k ) X P k k (k ) ( k ) 1 (k ) (k ) 1 2 ( k ) N ( , 2 ) X Example 10 |X | < |X | < 2 |X | < 3 P(| X | k ) 1 2 ( k ) P(| X | ) 1 2 (1) 1 2 0.1587 0.6826 P(| X | 2 ) 1 2(2) 1 2 0.0228 0.9544 P (| X | 3 ) 1 2(3) 1 2 0.0013 0.9974 Example 10 P(| X | k ) 1 2 ( k ) P(| X | ) 1 2 (1) 1 2 0.1587 0.6826 P(| X | 2 ) 1 2(2) 1 2 0.0228 0.9544 P (| X | 3 ) 1 2(3) 1 2 0.0013 0.9974 Chapter 4-1 Continuous Random Variables The Uniform Distributions The Uniform Distributions X ~ U (a, b), a b f(x) 1 , a xb pdf f ( x) ba 0 xa F ( x ) cdf b a 1 1 ba x a ax 1 b F(x) a xb bx x a b Summary The Exponential Distributions The Erlang Distributions The Gamma Distributions The Gaussian or Normal Distributions The Uniform Distributions