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Unformatted text preview: sum of r independent exponential random variables is the
scaled limit of the sum of r independent geometric random variables. That is exactly what we
just showed: The gamma distribution with parameters r and λ is the limit of a scaled negative
exponential random variable hSr , where Sr has the negative binomial distribution with parameters
r and p = λh. 3.6. LINEAR SCALING OF PDFS AND THE GAUSSIAN DISTRIBUTION 3.6
3.6.1 87 Linear scaling of pdfs and the Gaussian distribution
Scaling rule for pdfs Let X be a random variable with pdf fX and let Y = aX + b where a > 0. The pdf of Y is given
by the following scaling rule:
Y = aX + b ⇒ fY (v ) = fX v−b
a 1
.
a (3.3) We explain what the scaling rule means graphically, assuming a ≥ 1. The situation for 0 < a ≤ 1 is
similar. To obtain fY from fX , ﬁrst stretch the graph of fX horizontally by a factor a and shrink it
vertically by a factor a. That operation leaves the area under the graph equal to one, and produces
the pdf of aX. Then sh...
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 Spring '08
 Zahrn
 The Land

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