Lecture8 - Lecture 8: Conditional Probabilities; Random...

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Unformatted text preview: Lecture 8: Conditional Probabilities; Random Variables Chapter 5: Probability and Sampling Distributions 2/3/12 Lecture 8 1 Bayes Rule: Ex.13, (c) on Pg 209 For any two events A and B, P(A B) = P(A | B) P(B) = P(B | A) P(A) P(B) = P(B A) + P(B A ) = P(B | A) P(A) + P(B | A ) P(A ) If A and B are any events whose probabiliNes are not 0 or 1, then P( A B) P( B | A) P( A) P( A | B) = = P( B) P( B | A) P( A) + P( B | A' ) P( A' ) 2/3/12 Lecture 8 2 Quality Control A part is produced by 2 machines. A manufacturer is trying to idenNfy ways to reduce the # of defecNve parts. The present procedures produce 5% defecNves. AVer inspecNon, it was found that 40% defecNves and 15% of nondefecNves were produced on machine 1. Q: the probability of producing defecNve parts on machine 1? 2/3/12 Lecture 8 3 Quality Control A = a part is defecNve B = a part is produced by machine 1 Want to know P(A | B) = ? P(A) =.05, P(A) = .95. P(B|A)=.4, and P(B|A) = . 15. P( B | A) P( A) P( A | B) = P( B | A) P( A) + P( B | A' ) P( A' ) .4 .05 = = 12.3%. .4 .05 + .15 .95 2/3/12 Lecture 8 4 Lab Test A lab test yields 2 possible results: posi-ve or nega-ve. 99% of people with a parNcular disease will produce a posiNve result. But 2% of people without the disease will also produce a posiNve result. Suppose that .1% of the populaNon actually has the disease. Q: the probability that a person chosen at random will have the disease, given that the person's blood yields a posiNve result? 2/3/12 Lecture 8 5 Lab Test D = disease, + = posiNve test result. Want P(D | +) = ? P(D) =.001, P(D ) = 0.999. P(+ | D)=.99, and P(+ | D ) = .02. Applying Bayes Rule, P( + | D) P( D) P( D | +) = P( + | D) P( D) + P( + | D' ) P( D' ) .99 .001 .00099 = = 4.7%. .99 .001 + .02 .999 .02097 Lecture 8 6 2/3/12 Bayes rule 2/3/12 Lecture 8 7 Free-throws A BoilerMaker basketball player is a 80% free- throw shooter. Suppose he will make 20 free-throws during each pracNce. How many hits does he make on average during pracNce? 2/3/12 Lecture 8 8 Random Variables Experiment: A BoilerMaker basketball player makes 20 free throws during his pracNce. Define X: number of hits A random variable is a variable whose value is a numerical outcome of a random/chance experiment. 2/3/12 Lecture 8 9 Two types of random variables A discrete random variable has a finite number of possible values or an infinite sequence of countable real numbers. X: number of hits when trying 20 free throws. X: number of customers who arrive at the bank from 8:30 9:30AM Mon-Fri. E.g. Binomial, Poisson ... A con-nuous random variable takes all values in an interval of real numbers. X: the Nme it takes for a bulb to burn out. The values are not countable. Lecture 8 2/3/12 10 AVer Class ... Review the notes Finish Hw#3 due by 5pm, next Monday 2/3/12 Lecture 8 11 ...
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This note was uploaded on 02/06/2012 for the course STAT 350 taught by Professor Staff during the Spring '08 term at Purdue University-West Lafayette.

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