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UW-Madison ECON 310 - EconStats310 - September 19

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1 Econ 310 Professor Wallace September 19, 2013 Lecture: - Finish up Bayes’ rule examples (Section 6.4) - Random variables (Section 7.1) - Probability distributions for discrete random variables (Section 7.1) - Expectation, variance, and properties thereof (Section 7.1)2 Example 2: Data from the Office on Smoking and Health, Center for Disease Control and Prevention, indicate that 25 percent of adults who did not complete college smoke, compared with only 14 percent of college graduates that smoke. An individual is selected and random and it is determined that he is a smoker. Assuming that 33 percent of adults have graduated from college, what is the likelihood that the selected smoker is a college graduate? What do we want to know? ( | )P College S What do we know? -   0.33 0.67cP College P College   -  | 0.14P S College  -  | 0.25CP S College  Is Bayes’ Theorem Applicable? - What is our A event? - Can the sample space be partitioned to College and Collegec? o College and CollegeC are mutually exclusive o Together they comprise the sample space - Do we know  |P S College and want  |P College S?                    ||||0.14 0.33 0.2170.14 0.33 0.25 0.67CCP S College P CollegeP College SP S College P College P S College P College  3 Example 3 (Monty Hall Problem): Back in the 1970s there was a popular TV game show called Let’s Make a Deal. The show’s host, Monty Hall, would select audience members and offer them a small prize (say $1,000) which could be exchanged for a gamble. On the shows set are three curtains A, B, and C. Behind one curtain is a new car worth $25,000. Behind the other two curtains are barnyard animals or something else no regular person wants. The gamble is for the audience member to select a curtain and receive the prize behind the curtain. There is a twist: In an attempt to make things more interesting Monty always exposes a barnyard animal behind one of the doors not selected by the contestant and offers the contestant an opportunity to alter their curtain choice. Suppose that you are playing Let’s Make a Deal. You select curtain A and Monty exposes a goat behind curtain C. Should you switch your choice to curtain B? To determine the answer to this question we essentially want to compare two probabilities ( | ) ( )( | )()P showC prizeA P prizeAP prizeA showCP showC and ( | ) ( )( | )()P showC prizeB P priceAP prizeB showCP showC If ( | ) ( | )P prizeB showC P prizeA showC then you would be better of switching.4 Some facts: - 1( ) ( ) ( )3P prizeA P prizeB P prizeC   - 1( | )2P showC prizeA  - if your choice of A is winning then Monty randomizes what door to open between the remaining two. - ( | ) 1P showC prizeB  - if the prize is behind curtain B then Monty has to open curtain C. - ( | ) 0P showC prizeC  - Monty is never going to open the curtain with the prize behind it. Recall the objective is to calculate ( | ) ( )( | )()P showC prizeA P prizeAP prizeA showCP showC and ( | ) ( )( | )()P showC prizeB P priceAP prizeB showCP showC To do this we must first calculate ()P showC    ( ) ( | ) ( ) ( | ) ( ) ( | ) ( )1 1 1 1 1 1 02 3 3 3 2P showC P showC prizeA P prizeAP showC prizeB P prizeBP showC prizeC P prizeC                     5 Now we can calculate 11( | ) ( ) 123( | )1( ) 32P showC prizeA P prizeAP prizeA showCP showC         and  11( | ) ( ) 23( | )1( ) 32P showC prizeB P prizeBP prizeB showCP showC   So you double your probability of winning the prize by switching your curtain choice. The intuition behind this result is that Monty is always going to show the curtain opposite the prize when your first choice is not correct. Because your first choice is not correct more often than not you are better off switching.6 Probability Distributions Random variable - a numerical description of the outcome of an experiment. There are two types of random variables – (1) discrete random variables – can take on finite number or infinite integer sequence of values (2) continuous random variables – can take on any value in an interval or collection of intervals Illustration 1: The time that it takes to get to work in the morning is a continuous random variable. Illustration 2: The number of Bs that you will get in class this semester is a discrete random variable. Illustration 3: The sum of two dice is a discrete random variable.7 Discrete Random Variables Probability function (PF) ()fx - is a function that returns the probability of x for discrete random variables The probability function describes the probability distribution of a random variable. If you have the PF then you know the probability of observing any value of x. Probability functions are only relevant for discrete random variables Notes on notation: - Our text references probability functions using ()Px rather than ()fx - I will always reference random variables using capital letters and values of random variables using small letters. For example, X references a random variable and x a specific, but unspecified, value off that random variable. Using this notation ( ) ( )f x P X x Requirements for Probability Functions: (1) ( ) 0fx (2) ( ) 1fx Cumulative Distribution Function (CDF) ()Fx- is a function that returns the probability that a random variableXis less than or equal to a valuex. The distribution function has the same interpretation for discrete and continuous random variables. The CDF is also sometimes called the distribution function (DF). Requirements for CDFs (1) ( ) 0Fx everywhere the distribution is defined (2) ()Fx non-decreasing everywhere the distribution is defined. (3) ( ) 1Fx as x8 Example: Consider the probability distribution of the number of Bs you will get this semester Expected Value and Variance The expected value, or mean, of a random variable is a measure of central location. ( ) ( )xSE X x f x   In the formula above each value is


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UW-Madison ECON 310 - EconStats310 - September 19

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