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Christopher Dougherty EC220 - Introduction to econometrics (review chapter) Slideshow: exercise r.2 Original citation: Dougherty, C. (2012) EC220 - Introduction to econometrics (review chapter). [Teaching Resource] © 2012 The Author This version available at: Available in LSE Learning Resources Online: May 2012 This work is licensed under a Creative Commons Attribution-ShareAlike 3.0 License. This license allows the user to remix, tweak, and build upon the work even for commercial purposes, as long as the user credits the author and licenses their new creations under the identical terms.

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1 EXERCISE R.2 R.2*A random variable X is defined to be the larger of the numbers when two dice are thrown, or the number if they are the same. Find the probability distribution for X.

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2 red green Suppose that one die is red and the other green. EXERCISE R.2

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3 red green Then, for example, if the red die is 4 and the green one is 6, X is equal to 6.

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4 EXERCISE R.2 red green Similarly, if the red die is 2 and the green one is 5, X is equal to 5.

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5 EXERCISE R.2 red green The table shows all the possible outcomes.

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6 X123456X EXERCISE R.2 red green If you look at the table, you can see that X can be any of the numbers from 1 to 6.

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7 Xf123456Xf EXERCISE R.2 red green We will now define f, the frequencies associated with the possible values of X.

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8 Xf Xf EXERCISE R.2 red green For example, there are seven outcomes which make X equal to 4.

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9 Xf EXERCISE R.2 red green Similarly you can work out the frequencies for the other values of X.

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10 Xfp EXERCISE R.2 red green Finally we will derive the probability of obtaining each value of X.

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11 Xfp EXERCISE R.2 red green If there is 1/6 probability of obtaining each number on the red die, and the same on the green die, each outcome in the table will occur with 1/36 probability.

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12 Xfp 111/36 233/36 355/36 477/36 599/ /36 Hence to obtain the probabilities associated with the different values of X, we divide the frequencies by 36. EXERCISE R.2 red green

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Copyright Christopher Dougherty 1999–2006. This slideshow may be freely copied for personal use

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