transforms

# transforms - 1 Laplace Stieltjes Transforms Consider a...

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1 Laplace Stieltjes Transforms Consider a non-negative-valued continuous random variable X . The Laplace Stieltjes Transform (LST) of X is given by ˜ F X ( s ) = E [ e - sX ] . Therefore mathematically the LST can be written (and computed) as ˜ F X ( s ) = Z 0 e - sx dF X ( x ) = Z 0 e - sx f X ( x ) dx where f X ( x ) is the PDF of the random variable X . Example 1: If X exp( λ ), then ˜ F X ( s ) = λ λ + s . Example 2: If X Erlang ( k, λ ), then ˜ F X ( s ) = ± λ λ + s ² k . Example 3: If X Unif (0 , 1), then ˜ F X ( s ) = 1 - e - s s . Properties A few properties of LSTs are described below: 1. LST uniquely identiﬁes a distribution function. That means if you are given an LST, you can uniquely determine a CDF for the random variable. However, obtaining the CDF or PDF from the LST is not easy except under three circumstances: when only numerical inversion is needed, when the LST can be written as partial fractions of the form given in one of the examples above, or when the corresponding LT (converting LST to LT will be discussed later) can be inverted. 2. Moments of the random variables can be directly computed using

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## This note was uploaded on 09/18/2011 for the course ISEN 609 taught by Professor Klutke during the Spring '08 term at Texas A&M.

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transforms - 1 Laplace Stieltjes Transforms Consider a...

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