It is also called the n 0 1 distribution the cdf of

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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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