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Unformatted text preview: next formula. If a < b, then {X b} = {X a} [ {a < X b} with the
two sets on the righthand side disjoint so
P (X b) = P (X a) + P (a < X b)
or, rearranging,
P (a < X b) = P (X b) P (X a) = F (b) F (a) (A.10) The last formula is valid for any random variable. When X has density function
f , it says that
Zb
f (x) dx = F (b) F (a)
a i.e., the integral can be evaluated by taking the di↵erence of the antiderivative
at the two endpoints.
To see what distribution functions look like, and to explain the use of (A.10),
we return to our examples. 213 A.2. RANDOM VARIABLES, DISTRIBUTIONS
Example A.14. Uniform distribution. f (x) = 1/(b a) for a < x < b.
8
>0
xa
<
F (x) = (x a)/(b a) a x b
>
:
1
xb
To check this, note that P (a < X < b) = 1 so P (X x) = 1 when x
P (X x) = 0 when x a. For a x b we compute
Zx
Zx
1
xa
P (X x) =
f (y ) dy =
dy =
ba
ba
1
a b and In the most important special case a = 0, b = 1 we have F (x) = x for 0 x 1.
Example A.15. Exponential distribution. f (x) = e
(
0
x0
F (x) =
1exx0 x for x 0. The ﬁrs...
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This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell University (Engineering School).
 Spring '10
 DURRETT
 The Land

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