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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. CONTINUOUSTYPE 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.
 Spring '08
 Zahrn
 The Land

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