B1 45730 13oct08

# B1 45730 13oct08 - 45 730 Probability Decision Making...

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45 730 Probability & Decision Making (10/13/08) Topics for today’s class: •Rev iew • Fill in online evaluations • Solve parts of the sample midterm

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Overview of Topics • Basic Probability Concepts • Discrete Random Variables • Decision Theory • Continuous Random Variables • Simulation Models
Formal Definition of Probability Probability P(A) is assigned to every event and obeys 1. If A is any event in the sample space S, then 2. Let A be an event in S, and let O i denote the basic outcomes. Then (the notation means that the summation is over all the basic outcomes in A) 3. P(S) = 1 1 P(A) 0 ) P(O P(A) A i =

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Probability Rules •T h e Complement rule : h e Addition rule: – The probability of the union of two events is 1 ) A P( P(A) i.e., = + P(A) 1 ) A P( = B) P(A P(B) P(A) B) P(A + =
Rules •P ( A U B ) = 1 P(A U B) c = 1 – P(A c B c ) • P(A U B) = P(A) + P(A c B) • Counting rules – Product of sample spaces (e.g. toss three dice) – Permutation (order matters – draw three cards, first card is Ace) – Combinations (order does not matter – draw three cards, exactly one Ace)

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Conditional probability What is the probability that event A occurs given that event B occurs (or conditional on event B occurring)? () ( ) | P AB PA B PB =
Multiplication Rule • Multiplication rule for two events A and B: • Similarly P(B) B) | P(A B) P(A = P(A) A) | P(B B) P(A =

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Statistical Independence • Two events are statistically independent if and only if: – Events A and B are independent when the probability of one event is not affected by the other event • If A and B are independent, then P(A) B) | P(A = P(B) P(A) B) P(A = P(B) A) | P(B = if P(B)>0 if P(A)>0
Bayes’ Theorem ) B )P( B | P(A B)P(B) | P(A B)P(B) | P(A P(A) B)P(B) | P(A A) | P(B + = =

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Suppose 10% of population develops lung cancer. Smokers’ probability of developing lung cancer is 50%. Smokers are 10% of population. Probability that a non-smoker develops cancer?
Bayes’ Theorem •w h e r e : E i = i th event of k mutually exclusive and collectively exhaustive events A = new event that might impact P(E i ) ) )P(E E | P(A ) )P(E E | P(A ) )P(E E | P(A ) )P(E E | P(A P(A) ) )P(E E | P(A A) | P(E k k 2 2 1 1 i i i i i + + + = = K

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Picture of Bayes’ Theorem
The Probability Distribution Function (pdf) of a discrete random variable As before, a discrete random variable X has a probability model that specifies: 1.The M (possibly infinite) possible outcomes that the random variable can take ( x 1 , x 2 , . .., x M ) 2.The probabilities of the different outcomes ( p 1 , p 2 , . .., p M ) where p i = P ( X = x i ) known as the probability distribution function of the random variable X Best visualized in a histogram

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Cumulative Distribution Function (cdf) •T h e cumulative distribution function , denoted F(x 0 ), shows the probability that X is less than or equal to x 0 • In other words, ) x P(X ) F(x 0 0 = = 0 x x 0 P(x) ) F(x
Expected value Definition: The expected value or mean of a discrete random variable is the weighted sum of the outcomes where the weights are the probabilities of the outcomes E ( X ) = P 1 × x 1 + P 2 × x 2 + . .. + P M × x M (sometimes denoted μ )

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Variance
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B1 45730 13oct08 - 45 730 Probability Decision Making...

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