1. Dr. Gary Blau, Sean HanMonday, Aug 13, 2007 Statistical Design of Experiments SECTION I Probability Theory Review

2. Monday, Aug 13, 2007 Dr. Gary Blau, Sean Han PROBABILITY For any real system or phenomenon, there will be a certain amount of variability associated with data generated by the system. Probability is the language used to characterize and interpret the variability of this data.

3. Monday, Aug 13, 2007 Dr. Gary Blau, Sean Han PROBABILITY Probability is the most widely used formalism for quantifying uncertainty. There are two views of probability the frequentist view and the subjective view.

4. Monday, Aug 13, 2007 Dr. Gary Blau, Sean Han PROBABILITY CONCEPTS Random experiment: an experiment in which the outcomes can be different even though the experiment is run under the same conditions. Sample space: The set of possible outcomes of a random experiment. Event: A specified subset of sample outcomes.

5. Monday, Aug 13, 2007 Dr. Gary Blau, Sean Han FREQUENTIST APPROACH Frequentist (Classical) View of Probability The probability of an event occurring in a particular trial is the frequency with which the event occurs in a long sequence of similar trials.

6. Monday, Aug 13, 2007 Dr. Gary Blau, Sean Han FREQUENTIST APPROACH If a random experiment can result in any one of N equally likely outcomes and if exactly n of these outcomes correspond to event A, then: P(A) = n/N The above definition is not very useful for real world decision making (outside games of chance) since it is not possible to conduct an experiment many times.

7. Monday, Aug 13, 2007 Dr. Gary Blau, Sean Han BAYESIAN APPROACH Personalistic View of Probability also known as Subjectivist or Bayesian Probability: P(A | I) The probability of an event, A, is the degree of belief that a person has that the event will occur, given all the relevant information, I, known to the person. In this case the probability is a function not only of the event but of the current state of knowledge of the decision maker.

8. Monday, Aug 13, 2007 Dr. Gary Blau, Sean Han BAYESIAN APPROACH Since different people have different information relative to an event, and these same people may acquire new information at different rates as time progresses in the Bayesian Approach, there is no such thing as “the” probability of an event.

9. Monday, Aug 13, 2007 Dr. Gary Blau, Sean Han AXIOMS OF PROBABILITY (These Axioms Apply to both Frequentist and Subjective Views of Probability) If EVENT A is defined on a sample space S then: (i)P(A) = Sum of the probabilities of all elements in A (ii) if A = S, then P(A) = P(S) = 1 (iii) 0 ≤ P(A) ≤ 1 (iv)if A c is the complement of A, then P(A c ) = 1 – P(A)

10. Monday, Aug 13, 2007 Dr. Gary Blau, Sean Han ADDITION RULES (i) P(AUB) = P(A) + P(B) – P(AB) (ii) If A and B are mutually exclusive (i.e. P(AB) = 0), then P(AUB) = P(A) +P(B) (iii) If A 1, A 2, A 3,…,A n are mutually exclusively and A 1 UA 2 UA 3 ….UA n = S this is said to be an exhaustive collection.

11. Monday, Aug 13, 2007 Dr. Gary Blau, Sean Han The probability of an event A occurring when it is known that some other event B has already occurred is called a conditional probability and denoted P(A|B) and reads “The probability that the event A occurs given that B has occurred”. It the joint occurrence of A and B is know, the conditional probability may be calculated from the relationship ： P(A|B) = P(AB)/P(B) CONDITIONAL PROBABILITY

12. Monday, Aug 13, 2007 Dr. Gary Blau, Sean Han MULTIPLICATION RULE Since: P(A|B) = P(AB)/P(B) And: P(B|A) = P(AB)/P(A) We have: P(AB) = P(A|B)P(B) = P(B|A)P(A) A AB B

13. Monday, Aug 13, 2007 Dr. Gary Blau, Sean Han TOTAL PROBABILITY RULE If A 1,A 2,A 3,…,A n are an exhaustive collection of sets, then: Or equivalently: A 1 A 2 B A 3 A 4 A 5

14. Monday, Aug 13, 2007 Dr. Gary Blau, Sean Han EXAMPLE FOR PROBABILITY RULES A manager is trying to determine the probability of successfully meeting a deadline for producing 1000 grams of a new active ingredient for clinical trials. He knows that the probability of success is conditional on the amount of support from management (manpower & facilities). Having been around for a while he can also estimate the probability of getting different levels of support from his management. Calculate the probability of successfully meeting the deadline.

15. Monday, Aug 13, 2007 Dr. Gary Blau, Sean Han EXAMPLE (DATA) Let A i be the event of the amount of management support $. Let B be the event of a successfully meeting the deadline Subjective Probabilities Management SupportP(A i )P(B|A i ) $ 1MM0.50.7 $ 2MM0.40.8 $ 3MM0.10.9

16. Monday, Aug 13, 2007 Dr. Gary Blau, Sean Han EXAMPLE (SOLUTION) P(B) = P(B|A i )P(A i ) = (.5)(.7) + (.4)(.8) + (.1)(.9) =.76 There is a 76% chance that the manager will be able to make the material on time.

17. Monday, Aug 13, 2007 Dr. Gary Blau, Sean Han INDEPENDENT EVENTS (i)Two Events If P(A|B) = P(A) and P(B|A) =P(B) for two events A and B (i.e. neither event is influenced by the outcome of the other being known), then A and B are said to be independent. Therefore: P(AB) = P(A)P(B) (ii)Multiple Events A 1, A 2, …, A n are independent events if and only if, for any subset A il, …, A ik of A 1, A 2, …, A n: P(A il A i2 …A ik ) = P(A il )P(A i2 )…P(A ik )

18. Monday, Aug 13, 2007 Dr. Gary Blau, Sean Han ON-SPEC PRODUCT EXAMPLE Based on historical data, the probability of an off-spec batch of material from a processing unit is.01. What is the probability of producing 10 successive batches of on-spec material?

19. Monday, Aug 13, 2007 Dr. Gary Blau, Sean Han ON-SPEC PRODUCT EXAMPLE (SOLUTION) Probability of on-spec product in a batch: P(A i ) = 1 -.01 =.99. Since the batches are independent, the probability of 10 successive on-spec batches is: P(A 1 A 2 …A 10 ) = P(A 1 )P(A 2 )…P(A 10 ) = (.99) 10

20. Monday, Aug 13, 2007 Dr. Gary Blau, Sean Han BAYES’ THEOREM Suppose the probability of an event A, P(A) is known before an experiment is conducted, this is called the prior probability, then the experiment is conducted and we wish to determine the “new” or updated probability of A. Let B be some event condition on A, the P(A|B) is called the likelihood function. Therefore by Bayesian theorem: P(A|B) = P(AB)/P(B) and P(B|A) = P(AB)/P(A), then: posterior likelihood * prior

21. Monday, Aug 13, 2007 Dr. Gary Blau, Sean Han DIAGNOSING A DISEASE EXAMPLE The analytical group in your development department has developed a new test for detecting a particular disease in humans. You wish to determine the probability that a person really has the disease if the test is positive. A is the event that an individual has the disease. B is the event that an individual tests positive.

22. Monday, Aug 13, 2007 Dr. Gary Blau, Sean Han DATA FOR EXAMPLE (Prior Information): Probability that an individual has a disease: P(A) =.01 Probability that an individual does not have the disease: P(A c ) =.99 (Likelihood) Probability of a positive test result if person has the disease: P(B|A) = 0.90 Probability of a positive test result even if person does not have the disease (False Positive): P(B|A c ) =.05

23. Monday, Aug 13, 2007 Dr. Gary Blau, Sean Han DETERMINE POSTERIOR PROBABILITY (Posterior, Calculated probability) P(A|B) = P(B|A)P(A) / P(B) = P(B|A)P(A) / (P(B|A)P(A)+P(B|A c )P(A c )) =.09*.01/(.09*.01+.05*.99) =.153 This is a rather amazing result that there is only a 15% chance of having the disease when test is positive even though there is a 90% chance of testing positive if one has the disease!

24. Monday, Aug 13, 2007 Dr. Gary Blau, Sean Han A Random Variable X assigns numerical values to the outcomes of a random experiment Each of these outcomes is considered an event and has an associated probability The formalism for describing the probabilities of all the outcomes is a probability distribution. RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS

25. Monday, Aug 13, 2007 Dr. Gary Blau, Sean Han DISCRETE PROBABILITY DISTRIBUTIONS When the number of outcomes from an experiment are countable the random variable has a discrete probability distribution p(x): p(x).2.1 10 20 30 40 50 x

26. Monday, Aug 13, 2007 Dr. Gary Blau, Sean Han CONTINUOUS PROBABILITY DISTRIBUTIONS When the number of outcomes from a random experiment is infinite (not countable for practical purposes) the random variable has a continuous probability distribution f(x). (i.e. probability density function). P(l ≤x≤ u) = area under curve from l to u. f(x) l u x

27. Monday, Aug 13, 2007 Dr. Gary Blau, Sean Han MOMENTS OF DISTRIBUTIONS Central Tendency (Mean) if x is discrete if x is continuous Scatter (Variance) if x is discrete if x is continuous

28. Monday, Aug 13, 2007 Dr. Gary Blau, Sean Han DISCRETE PROBABILITY DISTRIBUTION EXAMPLE The discrete probability distribution of X is given by: Calculate its mean and variance. xp(x) 00.25 10.5 20.25

29. Monday, Aug 13, 2007 Dr. Gary Blau, Sean Han EXAMPLE (SOLUTION) μ x = ∑ x p(x) = (0)(.25) + (1)(.5) + (2)(.25) = 1 σ = ∑ (x- µ X ) 2 p(x) = (0-1) 2 (.25) + (1-1) 2 (.5) + (2-1) 2 (.25) =.5

30. Monday, Aug 13, 2007 Dr. Gary Blau, Sean Han CONTINUOUS PROBABILITY DISTRIBUTION EXAMPLE In a controlled lab experiment, the error in measuring the reaction temperature in C is given by (1) Is f (x) a probability density function? (2) What is the probability the error is between 0 and 1?

31. Monday, Aug 13, 2007 Dr. Gary Blau, Sean Han EXAMPLE (SOLUTION) (1) So it is a probability density function. (2)

32. Monday, Aug 13, 2007 Dr. Gary Blau, Sean Han UNIFORM DISTRIBUTION The probability density function of a continuous uniform distribution is:

33. Monday, Aug 13, 2007 Dr. Gary Blau, Sean Han UNIFORM DISTRIBUTION

34. Monday, Aug 13, 2007 Dr. Gary Blau, Sean Han NORMAL DISTRIBUTION Normal (Gaussian) Distribution is the most frequently occurring distribution in entire field of statistics. Gauss found that such a distribution is represented by the probability density function: with:

35. Monday, Aug 13, 2007 Dr. Gary Blau, Sean Han STANDARD NORMAL DISTRIBUTION If the parameters of the normal distribution are: the defining random variable is called the standard normal random variable Z with probability density function The approximate values of cumulative distribution function are listed in Z table.

36. Monday, Aug 13, 2007 Dr. Gary Blau, Sean Han STANDARD NORMAL DISTRIBUTION The values of cumulative standard normal distribution function for the standard normal random variable Z can be used to find the corresponding possibilities for normal random variables X with E(X) = μ and V(X)= σ 2 using the following transformation to convert the distribution of X to Z:

37. Monday, Aug 13, 2007 Dr. Gary Blau, Sean Han NORMAL DISTRIBUTION EXAMPLE If X ~ N(50, 100) [Read “If the random variable X is distributed normally with mean 50 and variance 100], find the probability that P(42 ≤ X ≤ 62). P(42 ≤ X ≤ 62) = P((42-50)/10 ≤ Z ≤ (62-50)/10) = Z 1.2 – Z -.8 =.885 -.212 =.673

38. Monday, Aug 13, 2007 Dr. Gary Blau, Sean Han TRIANGULAR DISTRIBUTION