CEE6430: Probabilistic Methods in Hydrosciences

LECTURE#2Refreshing Concepts on Probability

Event DefinitionClassical Vs. Experimental Definition of

ProbabilityIndependence/Dependence of Events

Mutually Exclusiveness of EventsBAYE’S Theorem in terms of Probabilities

Mr. BAYE’S Life-His ContributionLikelihood and Probability – the difference?

Random Variables and PDFsTrivia

Ideas for Class Projects

CEE6430: Probabilistic Methods in Hydrosciences

Event• (Random) Event - an outcome of a random experiment• Sample Space – all possible outcomes.Example 1: Tossing a coin 10 times – an experiment. (tossing israndom as we cannot predict outcome ‘definitively’)Sample space ?– Head and TailGetting Head 3 times ? – a random event.

Example 2: Predicting Avg. Rainrate for tomorrow (assume you are inKey West Florida, not Arizona!)0 mm/hr <Rainrate< infinite mm/hr (Statistical Sample Space)0 mm/hr < Rainrate< 1500 mm/hr (Realistic/Hydrologic Sample Space)A Random Event- Predicting an average Rainrate between 10 and 20mm/hr.

CEE6430: Probabilistic Methods in Hydrosciences

Probability

• Refer to Lecture Note #2 (First Page)• Probability of an Event: (Intuitively) the Ratio of the times

the Event occurred to number of times N, the experiment was attempted (N goes to infinity).

• Probability can sometimes be deduced logically (e.g. for a ‘fair coin’, chances of getting a Head is 50%)

• Logical deduction often doesn’t work in Hydroscience

Example: Say we want to predict the probability that the river stage at Caney Fork river will exceed 15 ft during a thunderstorm. Is the problem as straight forward as tossing a coin?

CEE6430: Probabilistic Methods in Hydrosciences

Independence/Dependence

• Mutually Independent Events – Occurrence of events bears no relation to another.

• Mutually Exclusive Events – Occurrence of one event precludes the other. A binary concept.

• In Nature, there are many M.E events – Rain/No-rain; Drought/Flood; Day/Night etc…

• The Probability Laws for Independence and mutually-exclusiveness – look at Lecture Note#2 (page 2).

CEE6430: Probabilistic Methods in Hydrosciences

Conditional Probability

• Conditional Probability – In engineering, many problems are formulated based on the assumption that an event has occurred. E.g. For a dam that has not failed in 50 years, what is its probability not to fail the next 25 years? We need Conditional Probability to answer this question.

• Bayes’ Theorem• Derivation of Bayes’ Theorem

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CEE6430: Probabilistic Methods in Hydrosciences

Bayes’ Theorem – Quizz#1

• Relax - not due today! (Due Next Class)• A city has a uniformly gridded water distribution system (20 km wide

X10 km long). Pressures and flowrates are uniform everywhere. Assume Cartesian coordinate system the bottom left corner as origin. Show on your system, the following events (loss in region):

A= (water loss in region bounded by 0<x<6km, 0<y<3km)

B=(water loss in region bounded by 4<x<10km, 2<y<6km)

Assume probability of loss proportional to affected area.

What is the ‘prior’ probability of a loss in region A?

What is the ‘prior’ probability of a loss in region B?

If there is a loss in region B, what is the probability that it is also in

region A?

CEE6430: Probabilistic Methods in Hydrosciences

How indebted are we to Mr. Bayes?

• Mr. Bayes (actually his theorem) has facilitated combining different kinds of information and updating knowledge based on newly acquired information. It can be used to incorporate additional observations to improve apriori estimates of probability of

events. • 1702-1761. Mr. Bayes’ Theorem got

acceptance/publication after his death.

• In essence, Bayes's Theorem is a simple mathematical formula used for calculating

conditional probabilities. Now used in almost all fields of science.

Looks like John Wayne?

CEE6430: Probabilistic Methods in Hydrosciences

Recap

• Solving Trivia#2 (Lecture Note#2, page 4)

• Certainty Vs. Uncertainty

• What’s the difference Likelihood Vs. Probability?

• Optional reading assignment – Chapters 1 and 2 of ‘Introduction to the Theory of Statistics’ (Mood).

CEE6430: Probabilistic Methods in Hydrosciences

Random Variables and PDFs(Lecture Note#2 Pages 4-7)

• Cumulative Probability Distribution Function (CDF)

• Probability Density Function (PDF)

• Joint Probability Distribution Function

• Distribution and Density?

• Conditional CDF, PDF

• Bayes’ Theorem in Continuous form.

CEE6430: Probabilistic Methods in Hydrosciences

Class Project

• Discussion of Ideas.