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Unformatted text preview: t this is a probability function we recall
ex = 1
X xk k=0 k! (A.7) so the proposed probabilities are nonnegative and sum to 1.
In many situations random variables can take any value on the real line or
in a certain subset of the real line. For concrete examples, consider the height
or weight of a person chosen at random or the time it takes a person to drive
from Los Angeles to San Francisco. A random variable X is said to have a
continuous distribution with density function f if for all a b we have
P (a X b) =
f (x) dx
a Geometrically, P (a X b) is the area under the curve f between a and b.
In order for P (a X b) to be nonnegative for all a and b and for
P ( 1 < X < 1) = 1 we must have
f (x) 0 and
f (x) dx = 1
1 Any function f that satisﬁes (A.9) is said to be a density function. We will
now deﬁne three of the most important density functions. 212 APPENDIX A. REVIEW OF PROBABILITY Example A.11. Uniform distribution on (a,b).
1/(b a) a < x < b
f (x) =
The idea here is that we are picking a value “at random” from (a, b). That is,
values outside the interval are impossible, and all those inside have the same
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