An appropriate function g is given by the inverse

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Unformatted text preview: pressed in radians, let (θ mod 2π ) denote the equivalent angle in the interval [0, 2π ]. Thus, (θ mod 2π ) is equal to θ + 2πn, where the integer n is such that 0 ≤ θ + 2πn < 2π . ˜ Let Θ be uniformly distributed over [0, 2π ], let h be a constant, and let Θ = ((Θ + h) mod 2π ). Find the distribution of Θ. ˜ Solution: By its definition, Θ takes values in the interval [0, 2π ], so fix c with 0 ≤ c ≤ 2π and ˜ ≤ c}. Since h can be replaced by (h mod 2π ) if necessary, we can assume without seek to find P {Θ loss of generality that 0 ≤ h < 2π. Then Θ = g (Θ), where the function g (u) = ((u + h) mod 2π ) is graphed in Figure 3.18. Two cases are somewhat different: The first, shown in Figure 3.18(a), is that 0 ≤ c ≤ h. In this case, Θ ≤ c if Θ is in the interval [2π − h, 2π − h + c], of length c. Therefore, ˜ in this case, P {Θ ≤ c} = 2c . The other case is that h < c ≤ 2π, shown in Figure 3.18(b). In this π case, Θ ≤ c i...
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