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 j k ω e f ( ) d continuous 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 ( λ) Let ~ Ep. () λt f () t λe u t λ M () t t < λ λ t λ Φ ( ω) defined for all ω λ Compare with original FT ( c ) e λt () u t F λ > λ+ e λt () u t λ CF of Gaussian pdf ( σ ) Let ~ N m,. M () t e Φ ( ω) e σ mt+ t σ jmω ω 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 F f d j ( τ) τ ( ω) + π δ( ω) Φ + Φ ( ω) π δ( ω)

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

5 5 esidual Life Inter-arrival Time with F ( ) Age esidual Life andom Observation Instance time Eample. Eponential inter-arrival time. ( λ ) is the inter-arrival time. Suppose Ep. What is? Two different, intuitive answers are possible. First, since and the random entry time can be anywhere within the interval we enter, λ. λ Secondly, since the Poisson arrival process is memoryless, the net arrival occurs after an eponential rv and thus. λ Which answer is right? Eample. Bernoulli inter-arrival time. is the inter-arrival time. Suppose with prob with prob Let be the length of the interval we enter randomly. What is P? [ ] Eample. In general, Compare and. is the inter-arrival time. Suppose with prob p with prob p Let be the length of the interval we enter randomly. p with prob p + p p with prob p + p

6 6 We can see that, in general for any distribution of, [ ] P[ ] [ ] f P P i i i for discrete for continuous Theorem. esidual Life n+ Let be the inter-arrival time with cdf F. F r f r for r n ( n+ ) Proof In particular, ( + C ) Let denote the random variable that measures the length of the interval we enter randomly. Write [ ] f ( ) Δ P 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 Now consider the joint pdf f ( r, )., f (, r ) f () f ( r ), f noting f ( r ) provided r ( ) f ( ) r

7 7 f () r f (, r ) d r, ( ) d F ( r) r eq. r is the pdf of the residual life. r f ( r ) Taking T (Positive Fourier Transform) of eq. r and using eq. c, Φ ( ω) Differentiating eq. r, { u( r) } F ( r) ( ω) { } + πδ ( ω ) ( ω ) πδ ( ω ) Φ + Φ j ω ( r) () Φ ( ω) j () ( ω) ω ( ω) Φ Φ ω () ωφ ω +Φ ω j ω ( r) ( m) m m ecall Φ () j for any. From eq.. r, () () ωφ () lim ω +Φ ω Φ j ω ω using L'Hospital's rule () ( ) ( ω) () Φ ω ωφ ω +Φ ω lim j ω ω Φ lim j 4 ω ( r )

8 8 Eq.4 r says j j or j ( r5) A more popular form of eq.r5 is + + σ ( C ) ( r6) which is the mean of the residual life. For a deterministic, periodic arrival process, C and thus. For an eponential, C and thus. Homework. Derive the nd moment of Show. Note. + ( C ) + C We note and if and only if C. For an eponential, C and thus. Now we understand why we wait at a bus stop longer during a busy hour than non-busy.

9 9 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 of. σ C σ 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

10 A simple intuitive approach is as follows: 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 Eample. Bernoulli inter-arrival time. Consider the following two cases. In the first case, is either or with equal probabilities. In the second case, alternates between and. Compare the resulting residual lives. Same or different? 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.