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Unformatted text preview: 1 − Φ(1)))K =P = (2Φ(1) − 1)K = (0.6826)K,
so K = 1/0.6826 ≈ 1.46. The same reasoning can be used to ﬁnd FX . For 0 ≤ v ≤ 4,
FX (v ) = P {0 ≤ X ≤ v } = P {0 ≤ Z ≤ v }K = P
= Φ v−2
2 − Φ(−1) K = Φ v−2
2 −1 ≤ Z −2
v−2
≤
2
2 K − 0.1587 K. Thus, 0 FX (v ) =
Φ 1 v −2
2 − 0.1587 K if v ≤ 0
if 0 < v ≤ 4
if v ≥ 4. Finally, as for ﬁnding E [X ], since the pdf fX (u) is symmetric about the point u = 2, and has
bounded support (so the mean exists), it follows that E [X ] = 2. 3.6. LINEAR SCALING OF PDFS AND THE GAUSSIAN DISTRIBUTION 3.6.3 93 The central limit theorem and the Gaussian approximation The Gaussian distribution arises frequently in practice, because of the phenomenon known as
the central limit theorem (CLT), and the associated Gaussian approximation. There are many
mathematical formulations of the CLT which diﬀer in various details, but the main idea is the
following: If many independent random variables are added together, and if each of them is small
in magnitude compared to the sum, then the sum h...
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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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