Lecture Notes
Chapter Five: Continuous Random Variables
Randall Miller
1 
Page
1.
Continuous Probability Distributions
The graphical form of the probability distribution for a continuous random variable
x
is a smooth
curve.
This curve, a function of
x
, is denoted by the symbol
( )
fx
and is variously called a
probability distribution function
, a
frequency function
, or a
probability distribution.
The areas under a probability distribution correspond to probabilities for
x
.
Let
A
be the area
beneath the curve between two points, say,
a
and
b
.
This area is the probability that
x
assumes a
value between
a
and
b
( )
axb
<<
.
Because there is no area over a single point, say,
x
=
a
, it
follows that the probability associated with a particular value of
x
is equal to 0; that is,
( )
0
Px a
=
=
, and hence
( ) ( )
Pa x b Pa x b
<< =
≤≤
.
Therefore, it does not matter whether the
endpoints are included, it will not affect the probability.
It follows that the total area under the
probability distribution assigned to the set of all values of
x
– should equal 1.
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 Fall '08
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 Statistics, Probability

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