Unformatted text preview: bability density. Note that the last property implies f (x) = c for a < x < b.
In this case the integral is c(b a), so we must pick c = 1/(b a).
Example A.12. Exponential distribution.
(
exx0
f (x) =
0
otherwise
Here
that > 0 is a parameter. To check that this is a density function, we note
Z1
1
e x dx = e x 0 = 0 ( 1) = 1
0 In a ﬁrst course in probability, the next example is the star of the show.
However, it will have only a minor role here.
Example A.13. Normal distribution.
f (x) = (2⇡ ) 1 /2 e x2 /2 Since there is no closed form expression for the antiderivative of f , it takes
some ingenuity to check that this is a probability density. Those details are not
important here, so we will ignore them.
Any random variable (discrete, continuous, or in between) has a distribution function deﬁned by F (x) = P (X x). If X has a density function f (x)
then
Zx
F (x) = P ( 1 < X x) = f (y ) dy 1 That is, F is an antiderivative of f .
One of the reasons for computing the distribution function is explained by
the...
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This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell.
 Spring '10
 DURRETT
 The Land

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