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Understanding Return Periods and Joint Distributions in Poisson Processes

This tutorial explores the concept of return periods in Poisson processes, focusing on a practical example where the exceedance probability within a building's lifetime is given. We analyze random variables, compare Poisson and binomial distributions, and discuss joint and marginal probability density functions (PDFs) for continuous random variables. The tutorial includes calculations for marginal and conditional distributions using a bivariate normal distribution and an example involving daily water levels in two reservoirs. Practical exercises are provided for deeper understanding.

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Understanding Return Periods and Joint Distributions in Poisson Processes

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  1. CIVL 181 Tutorial 5 Return period Poisson process Multiple random variables

  2. A question on return period If P (exceedence within the life time of the building, i.e., 10 years) = 0.1 Return period T = 100 years?

  3. Poisson process 1. The r.v. is continuous or discrete? 2. What is the relation between Poisson and binomial? 3. v / vt?

  4. Comparison of two families of occurrence models

  5. Joint and marginal PDF of continuous R.V.s marginal PDF fX (x) marginal PDF fY (y) x=a fX (a) = Area fX,Y (x, y=b) Surface = fX,Y (x,y) y =b Joint PDF fY (b) = Area Conditional PDF of Y given x=a fY|X(y|x =a) fX,Y (x=a, y)

  6. a) Calculate probability

  7. b) Derive marginal distribution

  8. c) Conditional distribution

  9. Example: Bivariate normal distribution (3.55) A formal def of bivariate normal distribution is: also by arithmetic we can rewrite as Find P(4 <Y< 6) if fX(x) is N (3,1), fY(y) is N (4,2) = 0.2 when x = 3, 3.5, 4

  10. Take x = 3.5 as example Compare to (Double integral!)

  11. (Take x = 3.5 as example)

  12. (Take x = 3.5 as example,) Knowing fY|X(y|x) = N (4.2, 1.95) P(X = 3.5, 4 <Y< 6) = 0.361 Try X = 4, X = 3 as exercise 4.2 1.95

  13. Ex 3.58 The daily water levels (normalized to respective full condition) of 2 reservoirs A and B are denoted by two r.v. X and Y have the following joint PDF:

  14. (a) Determine the marginal density function of daily water level for reservoir A

  15. (b) If reservoir A is half full on a given day, what is the chance that water level will be more than half full?

  16. (c) Is there any statistical correlation between the water levels in the two reservoirs?

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