This result does not depend on the distribution of

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Unformatted text preview: because cos−1 (y ) has derivative, −(1 − y 2 )− 2 , fB (c) = √1 π a2 −c2 0 |c| < a . |c| > a A sketch of the density is given in Figure 3.16. fB −a 0 a Figure 3.16: The pdf of the effective speed in a uniformly distributed direction. Example 3.8.6 Suppose Y = tan(Θ), as illustrated in Figure 3.17, where Θ is uniformly distributed over the interval (− π , π ) . Find the pdf of Y . 22 104 CHAPTER 3. CONTINUOUS-TYPE RANDOM VARIABLES 1 ! 0 Y Figure 3.17: A horizontal line, a fixed point at unit distance, and a line through the point with random direction. Solution: The function tan(θ) increases from −∞ to ∞ over the interval (− π , π ), so the support 22 of fY is the entire real line. For any real c, FY (c) = P {Y ≤ c} = P {tan(Θ) ≤ c} = P {Θ ≤ tan−1 (c)} = tan−1 (c) + π π 2 . Differentiating the CDF with respect to c yields that Y has the Cauchy distribution, with pdf: fY (c) = 1 π (1 + c2 ) − ∞ < c < ∞. Example 3.8.7 Given an angle θ ex...
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This note was uploaded on 02/09/2014 for the course ISYE 2027 taught by Professor Zahrn during the Spring '08 term at Georgia Tech.

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