Understanding Laplace Transforms for Continuous Random Variables

The Laplace transform is introduced as a generating function for common continuous random variables, complementing the z-transform for discrete ones. By using the Laplace transform, complex evaluations become simplified, making it easy to analyze different types of transforms. The transform of a continuous random variable with a non-negative continuous probability density function is explained, highlighting that convergence is guaranteed for non-negative random variables.

• Laplace Transforms
• Continuous Random Variables
• Generating Functions
• Probability Computing
• Transform Analysis

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1. Chapter 11 Laplace Transforms "Introduction to Probability for Computing", Harchol-Balter '24 1

2. There are different types of transforms Back in Chapter 6 we covered a type of generating function called the z-transform. The z-transform is particularly well suited to discrete, integer-valued random variables. In this chapter we introduce a new generating function called the Laplace transform, which is well suited to common continuous random variables. The structure of this chapter will closely mimic that of Chapter 6. 2 "Introduction to Probability for Computing", Harchol-Balter '24

3. Motivation Let ? ???(?) What is ? ?3? ? ?3= ?3 ?? ???? 0 Seems complicated to evaluate! The Laplace transform will make this very easy! 3 "Introduction to Probability for Computing", Harchol-Balter '24

4. The Laplace transform as an onion Onion represents Laplace transform of r.v. ? Lower moments are in the outer layers less effort/tears Higher moments are deeper inside more effort/tears 4 "Introduction to Probability for Computing", Harchol-Balter '24

5. Laplace transform of continuous r.v. Defn: Let ? be a non negative continuous r.v. with p.d.f. ??? . Then the Laplace transform of ? is ?(?) = ? ? ??= ? ????? ?? 0 Assume ? is a constant where ? 0. Note: The Laplace transform can be defined for any r.v., or even for just a function ? ? , where ? 0. However convergence is only guaranteed when ? is a non-negative r.v. and ? 0. 5 "Introduction to Probability for Computing", Harchol-Balter '24

6. Pop Quiz Defn: Let ? be a non negative continuous r.v. with p.d.f. ??? . Then the Laplace transform of ? is ?(?) = ? ? ??= ? ????? ?? 0 Assume ? is a constant where ? 0. Q: What is ? 0 ? A: ? 0 = ? ? 0 ?= 1 6 "Introduction to Probability for Computing", Harchol-Balter '24

7. Example of Onion Building ? ???(?) Create the onion! ? ? = ? ? ?? ?(?) = ? ? ??= ? ???? ???? 0 ? ????? ?? = 0 ? (?+?)??? = ? 0 ? = ? + ? 7 "Introduction to Probability for Computing", Harchol-Balter '24

8. Example of Onion Building ? = 3 Create the onion! ? ? = ? ? ?? ?(?) = ? ? ?? ? ????? ?? = = ?[? 3?] 0 = ? 3? 8 "Introduction to Probability for Computing", Harchol-Balter '24

9. Example of Onion Building ? ???????(?,?), where ?,? 0 Create the onion! ? ? = ? ? ?? ? 1 ?(?) = ? ? ??= ? ?? ? ??? ? ? ????? ?? = ? ? 1 1 0 ? ? ?? ? ?? = 9 "Introduction to Probability for Computing", Harchol-Balter '24

10. Convergence of Laplace transform Theorem 11.7: ? ? is bounded for any non negative continuous r.v. ?, assuming ? 0. ? ? 1, Proof: ? 0 ? ? ? 1, ? 0 ? ?? 1, ?,? 0 ? ? = ? ????? ?? 1 ??? ?? = 1 0 0 10 "Introduction to Probability for Computing", Harchol-Balter '24

11. Getting moments: Onion peeling Theorem 11.8: (Onion Peeling) Let ? be a non negative, continuous r.v. with p.d.f. ??? , ? 0. Then, ? ? s=0= ?[X] ? ? s=0= ?[X2] ? ? s=0= ?[X3] ? ? s=0= ?[X4] If can t evaluate at ? = 0, instead consider limit as ? 0 (use L Hospital s Rule). 11 "Introduction to Probability for Computing", Harchol-Balter '24

12. Proof of onion peeling theorem ??2 2! ??3 3! ??4 4! ? ??= 1 ?? + + (Taylor Series Expansion) ??2 2! ??3 3! ??4 4! ? ??? ? = ? ? ?? ? ? + ? ? ? ? + ? ? ??2 ??3 ? ??? ? ?? = ? ? ?? ?? ? ? ?? + ? ? ?? ? ? ?? + 2! 3! 0 0 0 0 0 ? ? = 1 ? ? ? +?2 2!? ?2 ?3 3!? ?3+?4 4!? ?4 ?5 5!? ?5+ 12 "Introduction to Probability for Computing", Harchol-Balter '24

13. Proof of onion peeling theorem ? ? = 1 ? ? ? +?2 2!? ?2 ?3 2!? ?3+?3 3!? ?3+?4 3!? ?4 ?4 4!? ?4 ?5 4!? ?5+?5 5!? ?5+?6 6!? ?6 ? ? = ? ? + ?? ?2 ?2 5!?[?6] ? 0 = ? ? ? ? = ? ?2 ?? ?3+?2 2!? ?4 ?3 3!? ?5+?4 4!? ?6 ? 0 = ?[?2] ? ? = ? ?3+ ?? ?4 ?2 2!? ?5+?3 3!? ?6 ? 0 = ?[?3] 13 "Introduction to Probability for Computing", Harchol-Balter '24

14. Example of onion peeling ? ? ? = ? + ?= ? ? + ? 1 ? ???(?) Q: Peel the onion to get ? ? , ?[?2],? ?3, ?[?4], ? ? =1 ? ? = ? ? + ? 2 ? ? ?2=2 ? ? = 2? ? + ? 3 ?2 ? ?3=3! ? ? = 3!? ? + ? 4 ?3 ? ??=?! ?? 14 "Introduction to Probability for Computing", Harchol-Balter '24

15. Linearity of Transforms Theorem 11.10: (Linearity) Let ? and ? be independent continuous r.v.s. Let ? = ? + ? Then the Laplace transform of ? is: ? ? = ? ? ? ? ? ? = ? ? ??= ?[? ? ?+?] Proof: = ?[? ?? ? ??] = ? ? ?? ? ? ?? = ? ? ? ? 15 "Introduction to Probability for Computing", Harchol-Balter '24

16. Conditioning with Transforms Theorem 11.11: Let ?, ?,and ? be continuous r.v.s. where ? = ? ? w.p. w.p. ? 1 ? Then, ?(?) = ? ?(?) + 1 ? ?(?) ?(?) = ? ? ?? Proof: = ? ? ??? = ?] ? + ? ? ??? = ?] (1 ?) = ?[? ??] ? + ?[? ??] (1 ?) = ? ?(?) + 1 ? ?(?) 16 "Introduction to Probability for Computing", Harchol-Balter '24

17. Conditioning Theorem 11.12: Let ? be a continuous r.v. and let ??be a continuous r.v. that dependes on ?. Let ??(?) denote the p.d.f. of ?. Then: ??? = ?=0 ??? ??? ?? ?= Proof: ??? = ? ? ??? = ? ? ???|? = ? ??? ?? ?=0 ?= ? ? ??? ??? ?? = ?=0 ??? ??? ?? = ?=0 17 "Introduction to Probability for Computing", Harchol-Balter '24