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Unformatted text preview: niformly distributed over [7, 13].
Therefore, P {X < 8.27} = 8.27−7 ≈ 0.211.
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(b) A random variable uniformly distributed on [a, b] has mean In some applications the distribution of a random variable seems nearly Gaussian, but the
random variable is also known to take values in some interval. One way to model the situation is
used in the following example.
Example 3.6.6 Suppose the random variable X has pdf (u − 2)2
K , √ exp − 2 2π
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fX (u) = 0, 0 ≤ u ≤ 4, otherwise. where K is a constant to be determined. Determine K , the CDF FX , and the mean E [X ].
Solution: Note that for u ∈ [0, 4], fX (u) = KfZ (u), where Z is a N (µ = 2, σ 2 = 4) random
variable. That is, fX is obtained by truncating fZ to the interval [0, 4] (i.e. setting it to zero outside
the interval) and then multiplying it by a constant K > 1 so fX integrates to one. Therefore,
∞ 1= 4 fZ (u)du K = P {0 ≤ Z ≤ 4} K fX (u)du =
−∞ 0 0−2
Z −2
4−2
≤
≤
K
2
2
2
Z −2
≤1 K
= P −1 ≤
2
= (Φ(1) − Φ(−1)) K = (Φ(1) − (...
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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 Institute of Technology.
 Spring '08
 Zahrn
 The Land

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