CIVL 181Tutorial 5 Return period Poisson process Multiple random variables

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

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

Bernoulli Sequence Poisson Process IntervalDiscreteContinuous No. of occurrenceBinomialPoisson Time to next occurrenceGeometricExponential Time to kth occurrenceNegative binomialGamma Comparison of two families of occurrence models

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

a) Calculate probability

b) Derive marginal distribution

c) Conditional distribution

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 f X (x) is N (3,1), f Y (y) is N (4,2) = 0.2 when x = 3, 3.5, 4

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

(Take x = 3.5 as example)

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

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:

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

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

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