1
ECE 514
Lecture 4
Properties of a PDF
±
Lemma:
1.
is a nonnegative function , i.e. ,
2.
integrates to 1.
Note
:
()
X
f
x
() 0
X
fx
≥
x
R
∀∈
X
f
x
1
X
f
xdx
=
∫
lim
( )
( )
1
y
XX
yy
F
yf
x
d
x
−∞
→∞
=
=
∫
±
Question:
How do we use PDF’s to compute
Probabilities?
Lemma
: Recalling that
We have the following:
Question:
Does
satisfy Axiom 3 of
Probability?
x
F
xf
y
d
y
=
∫
(
)
b
X
a
Pa X b
f xdx
+
+
<≤=
∫
(
)
b
X
a
+
−
≤≤=
∫
(
)
b
X
a
−
−
≤<=
∫
(
)
b
X
a
−
+
<<=
∫
(
)
X
a
Pa x
+
∞
<=
∫
X
f
∫
Conditional PDF
±
Consider a probability space
and a
random variable defined on
.
Definition:
The conditional probability density
function of an RV
, given some event A
such that
is defined as:
{, ,}
P
Ω
F
Ω
X
PA
≥
( )
X
X
dF
x A
fxA
dx
=
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Empirical Distribution
Ex:
Suppose you are measuring a voltage V
1
which
is supposed to be constant.
±
As a
sensed voltage, it feeds a regulator(filter)
whose output needs to maintain a constant
temperature to within a
(
for example)
Problem:
To design a regulator we will need the
statistics (variations of
) i.e.
which
we need to establish experimentally.
1
Va
=
1
10
D
1
V
1
ii
n
=+
1,.
..
iT
=
(Continued)
Question:
How would we go about that?
Start with Definition
:
Or:
()
(
)
X
P
Xxd
xP
Xx
fx
dx
≤
+−
≤
≅
(
)
X
f xdx Px X x dx
≅
≤≤+
±
i.e. Find the mass in that fractional
segment.
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 Fall '08
 Krim
 Laplace, Probability theory, dx, Δx Δx

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