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Unformatted text preview: J. Virtamo 38.3143 Queueing Theory / Continuous Distributions 1 CONTINUOUS DISTRIBUTIONS Laplace transform (LaplaceStieltjes transform) Definition The Laplace transform of a nonnegative random variable X ≥ 0 with the probability density function f ( x ) is defined as f * ( s ) = integraldisplay ∞ e st f ( t ) dt = E[ e sX ] = integraldisplay ∞ e st dF ( t ) also denoted as L X ( s ) • Mathematically it is the Laplace transform of the pdf function. • In dealing with continuous random variables the Laplace transform has the same role as the generating function has in the case of discrete random variables. – if X is a discrete integervalued ( ≥ 0) r.v., then f * ( s ) = G ( e s ) J. Virtamo 38.3143 Queueing Theory / Continuous Distributions 2 Laplace transform of a sum Let X and Y be independent random variables with Ltransforms f * X ( s ) and f * Y ( s ). f * X + Y ( s ) = E[ e s ( X + Y ) ] = E[ e sX e sY ] = E[ e sX ]E[ e sY ] (independence) = f * X ( s ) f * Y ( s ) f * X + Y ( s ) = f * X ( s ) f * Y ( s ) J. Virtamo 38.3143 Queueing Theory / Continuous Distributions 3 Calculating moments with the aid of Laplace transform By derivation one sees f *prime ( s ) = d ds E[ e sX ] = E[ Xe sX ] Similarly, the n th derivative is f * ( n ) ( s ) = d n ds n E[ e sX ] = E[( X ) n e sX ] Evaluating these at s = 0 one gets E[ X ] = f * prime (0) E[ X 2 ] = + f * primeprime (0) . . . E[ X n ] = ( 1) n f * ( n ) (0) J. Virtamo 38.3143 Queueing Theory / Continuous Distributions 4 Laplace transform of a random sum Consider the random sum Y = X 1 + ··· + X N where the X i are i.i.d. with the common Ltransform f * X ( s ) and N ≥ 0 is a integervalued r.v. with the generating function G N ( z ). f * Y ( s ) = E[ e sY ] = E[E bracketleftbigg e sY  N bracketrightbigg ] (outer expectation with respect to variations of N ) = E[E bracketleftbigg e s ( X 1 + ··· + X N )  N bracketrightbigg ] (in the inner expectation N is fixed) = E[E[ e s ( X 1 ) ] ··· E[ e s ( X N ) ]] (independence) = E[( f * X ( s )) N ] = G N ( f * X ( s )) (by the definition E[ z N ] = G N ( z )) J. Virtamo 38.3143 Queueing Theory / Continuous Distributions 5 Laplace transform and the method of collective marks We give for the Laplace transform f * ( s ) = E[ e sX ] , X ≥ , the following Interpretation : Think of X as representing the length of an interval. Let this interval be subject to a Poissonian marking process with intensity s . Then the Laplace transform f * ( s ) is the probability that there are no marks in the interval. P { X has no marks } = E[P { X has no marks  X } ] (total probability) = E[P { the number of events in the interval X is 0  X } ] = E[ e sX ] = f * ( s ) í î ì X intensiteetti s P { there are n events in the interval X  X } = ( sX ) n n !...
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This note was uploaded on 01/05/2010 for the course ECE 01 taught by Professor All during the Spring '09 term at Aarhus Universitet.
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