Download Refreshing Concepts on Probability

LECTURE#2
Refreshing Concepts on Probability
Event Definition
Classical Vs. Experimental Definition of
Probability
Independence/Dependence of Events
Mutually Exclusiveness of Events
BAYE’S Theorem in terms of Probabilities
Mr. BAYE’S Life-His Contribution
Likelihood and Probability – the difference?
Random Variables and PDFs
Trivia
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 is
random as we cannot predict outcome ‘definitively’)
Sample space ?– Head and Tail
Getting Head 3 times ? – a random event.
Example 2: Predicting Avg. Rainrate for tomorrow (assume you are in
Key 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 20
mm/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 mutuallyexclusiveness – 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
P( Ai / B) 
P( B / Ai ) P( Ai )
n
 P( B / Ai ) P( Ai )
i 1
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.
Looks like John Wayne?
• In essence, Bayes's Theorem is a simple
mathematical formula used for calculating
conditional probabilities. Now used in
almost all fields of science.
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.
CEE6430: Probabilistic Methods
in Hydrosciences