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Unformatted text preview: 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 . 14.30 Introduction to Statistical Methods in Economics Lecture Notes 7 Konrad Menzel February 26, 2009 1 Joint Distributions of 2 Random Variables X, Y (ctd.) 1.1 Continuous Random Variables If X and Y are continuous random variables defined over the same sample space S . The joint p.d.f. of ( X, Y ), f XY ( x, y ) is a function such that for any subset A of the ( x, y ) plane, P (( X, Y ) ∈ A ) = f XY ( x, y ) dxdy A As in the single-variable case, this density must satisfy f XY ( x, y ) ≥ 0 for each ( x, y ) ∈ R 2 and ∞ ∞ f XY ( x, y ) dxdy = 1 −∞ −∞ Note that • any single point has probability zero • any one-dimensional curve on the plane has probability zero Example 1 A UFO appears at a random location over Wyoming, which - ignoring the curvature of the Earth - can be described quite accurately as a rectangle of 276 times 375 miles. The position of the UFO is uniformly distributed over the entire state, and can be expressed as a random longitude X (ranging from -111 to -104 degrees) and latitude Y (with values between 41 and 45 degrees). This means that the joint density of the coordinates is given by 1 f XY ( x, y ) = 28 if − 111 ≤ x ≤ − 104 and 41 ≤ y ≤ 45 0 otherwise If the UFO can be seen from a distance of up to 40 miles, what is the probability that it can be seen from Casper, WY (which is roughly in the middle of the state)? Let’s look at the problem graphically: This suggests that the set of locations for which the UFO can be seen from Casper can be described as a circle with a 40-mile radius around Casper. Also, for the uniform density, the probability of the UFO showing up in a region A (i.e. the integral of a constant density over 1 F xy 1 x y (-111, 45) (-104, 45) (-104, 41) (-111, 41) 375 mi. 276 mi. Casper, WY Wyoming 40 mi. (x,y) Figure 1: The UFO at ( x, y ) can be seen Image by MIT OpenCourseWare. from Casper, WY A ) of the state is proportional to the area of A . Therefore, we don’t have to do any integration, but finding the probability reduces to a purely geometric exercise. We can calculate the probability as Area(”less than 40 miles from Casper”) 40 2 π P (”less than 40 miles from Casper”) = = ≈ 4 . 9% Area(”All of Wyoming”) 375 · 276 You should notice that for the uniform distribution, there is often no need to perform complicated inte- gration, but you may be able to treat everything as a purely geometric problem. Unlike in the last example, typically, there’s no way around integrating the density function in order to obtain probabilities, since any nonconstant density re-weights different regions in terms of probability mass. We’ll do this in the clean, systematic fashion in the following example: Example 2 Suppose you have 2 spark plugs in your lawn mower, and let...
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