B1 45730 13oct08

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

Info iconThis preview shows pages 1–17. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Overview of Topics • Basic Probability Concepts • Discrete Random Variables • Decision Theory • Continuous Random Variables • Simulation Models
Background image of page 2
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 =
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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 + =
Background image of page 4
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)
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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 =
Background image of page 6
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 =
Background image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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
Background image of page 8
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 + = =
Background image of page 9

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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?
Background image of page 10
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
Background image of page 11

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Picture of Bayes’ Theorem
Background image of page 12
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
Background image of page 13

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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
Background image of page 14
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 )
Background image of page 15

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Variance
Background image of page 16
Image of page 17
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

Page1 / 68

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

This preview shows document pages 1 - 17. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online