Probability Density Functions, Page 1
Probability Density Functions
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)
x1 x2 x3 ...
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 22
ii
dx dx
Px x x
⎛⎞
−<≤+
⎜⎟
⎝⎠
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 xi – dx/2 and xi + dx/2, the
integral under the PDF curve is the
probability that a measurement lies
within that range, as sketched.
x
f(x)
xi + dx/2
0.02
0.03
0
0.01
xi – dx/2
dx
xi
22
ii
dx dx
Px x x
⎛⎞
−<≤+
⎜⎟
⎝⎠
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 we choose a range of variable x, say between a and b.
The probability that a measurement lies
between a and b is simply the integral
under the PDF curve between a and b,
as sketched, where we define the
probability as
()(
xb
xa
Pa x b f xdx
=
=
<≤ =
∫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
Px fxdx
=∞
=−∞
−∞ < < ∞ = =
∫.
In other words, the probability that x lies between –∞ and +∞ is 100% (a fact that should be obvious,
since there are no other possibilities for real number x).
o Once we have defined the probability density function f(x), we leave the system of discrete random
variables and enter the system of continuous random variables, on which we make some more formal
definitions:
Expected value is defined in terms of the probability density function as the mean of all possible x
values in the continuous system. Namely,
() ()
expected value
E
xxfx
μ
∞
−∞
== =
∫dx
. In an ideal
situation in which f(x) exactly represents the population,
μ
is the mean of the entire population of x
values, and that is why it is called the “expected” value. It is therefore also called the population
mean. In general,
x
≠
μ
, but x
→
μ
when n is large, i.e., the sample mean approaches the
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