Transform Techniques - CF

Transcription

1 Transform Techniques - CF [eview] Moment Generating Function For a real t, the MGF of the random variable is t t M () t E[ e ] e Characteristic Function (CF) k t k For a real ω, the characteristic function of is e p ( ) discrete t Φ ( ω) Ee [ ] e k e f d continuous k e k p ( ) discrete k ω e f d continuous j Note. Compare the CF with the common definition of the Fourier trasnform: Φ ( ω) t j t e dt. f ω is the Fourier transform of the pdf with ω in place of ω. So we can make use of the properties of the Fourier transform. For clarity, we will use the following notation, Positive Fourier Trasnform: F ( ω) f j f Φ ( ω) ( ω) Φ Probability Generating Function (often called as z-transform) For a discrete non-negative integer random variable, the PGF is n [ ] n G ( z) Ez z z P n z: comple variable When is an integer rv, PGF is more convenient than MGF or CF. We can make use of the properties of the z-transform.

2 Properties of Characteristic Functions Find the moment ecall from the MGF t t t e + t + + +!! t t t t M () t E[ e ] e + t + + +!! m ( m) d m M () M ( t) m dt t With the characteristic function, e !! m ( m) d m m () ( ω) j m dω ω Φ ( ω) Ee [ ] e !! Φ Φ Can show two random variables have the same probability distribution Φ ( ω) ΦY( ω) f( u) fy( u) If two random variables and Y have the same characteristic function, and Y have the same probability distribution. Sum of Independent Vs Consider Z + Y. If and Y are independent random variables with Φ ( ω) and Φ ( ω) respectively, then Φ ( ω) Φ ( ω) Φ ( ω) Z Y Y

3 CF of Eponential pdf ( ) Let ~ Ep. Then () t f () t e u t M () t t < t Φ ( ω) defined for all ω Compare with original FT e t e > + F () { } u t ( c ) e t () u t CF of Gaussian pdf ( σ ) Let ~ N m,. Then M () t e Φ ( ω) e σ mt + t σ jmω ω

4 4 T of cdf From the properties of the original Fourier transform, F f F f d j F ( τ) τ ( ω) + π δ( ω) where δ ω is the unit impulse function. Therefore the Positive Fourier transform will result in However Φ F f d j ( τ) τ ( ω) + π δ( ω) Thus we have Φ + Φ ( ω) π δ( ω) always equals in our application, the pdf. F f d c ( τ) τ Φ ( ω) + πδ ( ω)

5 5 Eample. Fourier Transform of Eponential cdf ( ) Let ~ Ep. Then f e We can wrtite eq.c as where u ( ) F e c ( 4) F u e u c is the unit step function. Then the Positive Fourier transform of the cdf can be found from { } { F } { } { } From the Fourier tranform table,, taking the opposite sign of ω, Thus we have F u e u { u } + πδ ( ω) { e u } + πδ ω. ( c5) { F } Alternatively we can find from eq.c and c, { F } + πδ ( ω ) which is indentical to eq.c5

6 6 esidual Life Inter-arrival Time with F Age esidual Life andom Observation Instance time Eample. Eponential inter-arrival time. ( ) Let be the inter-arrival time. Suppose Ep. What is? Two differnet, intuitive answers are possible. Firstly, since and the random entry time can be anywhere within the interval we enter,. Secondly, since the Poisson arrival process is memoryless, the net arrival occurs after an eponential rv and thus. Which one is the right answer? Cannot be both. Eample. Bernoulli inter-arrival time. Let be the inter-arrival time. Suppose with prob with prob Let be the length of the interval we enter randomly. Find P.

7 7 Eample. In general, Compare and. Let be the inter-arrival time. Suppose with prob p with prob p Let be the length of the interval we enter randomly. Then p with prob p + p p with prob p + p We can see that, in general for any, P P i p i i f i i i for discrete for continuous Theorem. esidual Life Let be the inter-arrival time with cdf F. Then F r f r for r n n+ ( n+ ) Proof Let Then Write In particular, ( + C ) denote the random variable that measures the length of the interval we enter randomly. P f Δ f Δ o f k f for some constant k. o k is found from the requirement that f d should be k f d k k o f f o r

8 8 Now consider the joint pdf f ( r, )., f (, r ) f () f ( r ), f f r Then, f () r f (, r ) d r d F ( r) r The pdf of the residual life is r f noting f ( r ) for r ( ) F () r f r u r r Taking T (Positive Fourier Transform) of eq. r and using eq. c, Φ ( ω) { u( r) } F ( r) ( ω) { } + πδ ( ω) Φ ( ω) + πδ ( ω) Φ j ω ( r) Differentiating eq. r, () Φ ( ω) j () ( ω) ω ( ω) Φ Φ ω () ωφ ω +Φ ω j ω ( r)

9 9 ( m) m m ecall Φ () j for any. From eq.. r, () Φ () lim j ω () ωφ ω +Φ ω ω using L'Hospital's rule () ( ) ( ω) () Φ ω ωφ ω +Φ ω lim j ω ω Φ lim j 4 ω ( r ) Eq.4 r says j j or j ( r5) A more popular form of eq.r5 is + σ + ( C ) ( r6) For a deterministic, periodic arrival process, C and thus. For an eponential, C and thus. Homework. Derive the nd moment of Show. Homework. andom Interval is the length of the interval we randomly enter. Show

10 Note. + ( C ) + C For an eponential, C and thus. C We note and if and only if. Now we understand why we wait at a bus stop longer during a busy hour than non-busy. Eample. Bernoulli inter-arrival time. Suppose the inter-arrival time is either or with equal probabilities. Find the mean value and the pdf of the residual life. Solution Find the coefficient of variation. σ C σ 4 4 C Find the average residual life ( + C ) Find the distribution of the residual life F () r f r u( r) f ( ) + δ( ) δ F u ( ) + u ( ) F ( r) U(,) + U(,) f () r U(,) + U(,) b f ( r) r

11 An intuitive approach is simpler than the formal one. When we enter the arrival process at a random time, the interval we enter is with prob and is with prob. Therefore the residual life is U (,) with prob and is U (,) with prob. f () (,) r U + U(,) which is same as eq. b. U (,) U (, ) + r r Homework. Two periodic arrivals: Arrival Process : periodic with period Arrival Process : periodic with period The starting times of the two arrival processes are chosen randomly. source arrivals θ θ t source arrivals Find the mean value and the pdf of the residual life.