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Unformatted text preview: f Θ is in the union of intervals [2π − h, 2π ] ∪ [0, c − h], which has total length c. So,
˜
˜
˜
again, P {Θ ≤ c} = 2c . Therefore, in either case, P {Θ ≤ c} = 2c , so that Θ is itself uniformly
π
π
distributed over [0, 2π ].
Angles can be viewed as points on the unit circle in the plane. The result of this example is
that, if an angle is uniformly distributed on the unit circle, then the angle plus a constant is also
uniformly distributed over the unit circle. 3.8. FUNCTIONS OF A RANDOM VARIABLE (a) 105 (b)
2! 2! g(u)
g(u) c
h h
c
2!
0 2 ! !h 2!!h+c u u
0 c!h 2 ! !h 2! Figure 3.18: The function g such that Θ = g (Θ). Example 3.8.8 Express the pdf of X  in terms of the pdf of X, for an arbitrary continuoustype
random variable X. Draw a sketch and give a geometric interpretation of the solution.
Solution: We seek the pdf of Y, where Y = g (X ) and g (u) = u. The variable Y takes nonnegative
values, and for c ≥ 0, FY (c) = P {Y ≤ c} = P {−c ≤ X ≤ c} = FX (c) − FX...
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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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