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MIT OpenCourseWare http://ocw.mit.edu 14.30 Introduction to Statistical Methods in Economics Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .

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Problem Set #3 14.30 - Intro. to Statistical Methods in Economics Instructor: Konrad Menzel Due: Tuesday, March 3, 2009 Question One 1. Write down the definition of a cumulative distribution function (CDF). Explain what it means in words, perhaps using an example. Solution to (1): One definition of the CDF is f (.) : R H [0,1] where f (x) - PT(X 5 x). The CDF tells us the cumulative probability up to particular point of the ordered support of the random variable, X. What this means is that we can know what the chances are that something less than or equal to (or to the left, depending on how you wish to interpret the ordering) an outcome, x, occurs. 2. Verify whether the following function is a valid CDF. If yes, draw a graph of the corresponding PDF. x F x (x) 1 3 4 1 2 1 5 1 2 3 4 5 - - Solution to (2): The function is in fact a valid CDF. It is bounded below by zero and above by one. It also satisfies the left and right limit conditions as lim,,-, Fx(x) = 0 and lim,,, Fx(x) = 1. However, it is a mixture random variable where it has a continuous distribution and a mass point at 2. The PDF by MIT OpenCourseWare. Image
is the following equation: 3. Verify that the following function is a valid PDF and draw the corresponding CDF a Solution to This function is, in fact, a valid PDF. It is po F x (x) 1 2 3 4 5 6 x 1 3 sitive everywhere and integrates to 1 (the triangle Image by MIT OpenCourseWare. has area of and the interv \$ al from 5 to 6 has area of which together sum to 1). The CDF is straightforward. I will write the PDF and then CDF down analytically first, to make for easier integration: Drawing this curve is relatively straightforward, at least if you pay little attention to detail as I am not a graphic designer:

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