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Unformatted text preview: 9999915. The Φ and Q functions are available on many programable calculators and
on Internet websites.1 Some numerical values of these functions are given in Tables 6.1 and 6.2, in
the appendix.
Let µ be any number and σ > 0. If X is a standard Gaussian random variable, and Y = σX + µ,
then Y is a N (µ, σ 2 ) random variable. Indeed,
1
u2
fX (u) = √ exp −
2
2π , so by the scaling rule (3.3),
1
v−µ
fX
σ
σ
(v − µ)2
1
√
exp −
2σ 2
2πσ fY (v ) =
= , so fY is indeed the N (µ, σ 2 ) pdf. Graphically, this means that the N (µ, σ 2 ) pdf can be obtained
from the standard normal pdf by stretching it horizontally by a factor σ, shrinking it vertically by
a factor σ, and sliding it over by µ.
Working in the other direction, if Y is a N (µ, σ 2 ) random variable, then the standardized version
−
of Y , namely X = Y σ µ , is a standard normal random variable. Graphically, this means that the
standard normal pdf can be obtained from a N (µ, σ 2 ) pdf by sliding it over by µ (so it becomes
c...
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 Spring '08
 Zahrn
 The Land

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