Probability Density Functions
Author: John M. Cimbala, Penn State University
Latest revision: 04 September 2007
Probability Density Functions
•
Probability density function
– In simple terms, a
probability density function
(
PDF
) is constructed by
drawing a smooth curve fit through the
vertically normalized histogram as
sketched. You can think of a PDF as
the
smooth limit of a vertically normalized
histogram
if there were millions of
measurements and a huge number of bins.
o
The main difference between a
histogram and a PDF is that a
histogram involves
discrete data
(individual bins or classes), whereas a PDF involves
continuous data
(a smooth curve).
x
f
(
x
)
x
1
x
2
x
3
...
0.02
0.03
0
0.01
o
Mathematically,
f
(
x
) is defined as
()
22
ii
i
dx
dx
Px
x x
fx
dx
⎛⎞
−<
≤
+
⎜⎟
⎝⎠
=
, where
dx
dx
≤
+
represents
the
probability that variable
x
lies in the given range
, and
f
(
x
) is
the probability density
function (PDF)
. In other words, for the
given infinitesimal range of width
dx
between
x
i
–
dx
/2 and
x
i
+
dx
/2,
the
integral under the PDF curve is the
probability that a measurement lies
within that range
, as sketched.
x
f
(
x
)
x
i
+
dx
/2
0.02
0.03
0
0.01
x
i
–
/2
x
i
≤
+
o
As shown in the sketch, this probability
is equal to the
area
(shaded blue region)
under the
f
(
x
) curve – i.e., the
integral
under the PDF
over the specified
infinitesimal range of width
dx
.
o
The usefulness of the PDF is as follows: Suppose one chooses a range of variable
x
, say between
a
and
b
.
The probability that a measurement l
between
a
and
b
is simply the integ
under the PDF curve between
a
and
b
,
as sketched, where we define the
probability as
ies
ral
(
xb
xa
Pa x b
f xd
x
=
=
<≤ =
∫
x
f
(
x
)
b
0.02
0.03
0
0.01
a
P
(
a
<
x
≤
b
)
)
o
If
a
→
–
∞
and
b
→
+
∞
, the probability
must equal 1 (100%), i.e.,
(
)
1
x
x
f
x
d
x
=∞
=−∞
−∞ <
< ∞ =
=
∫
.
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 Spring '08
 staff
 probability density function

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