This preview shows pages 1–6. Sign up to view the full content.
Cumulative Distribution Function of a Random Variable
Definition:
()
[
]
[{ :
( )
}]
X
F x
PX x
P s
Xs x
=
≤
=
−∞≤
≤
Other terms are cdf and Probability Distribution Function (PDF).
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document Example – Uniform Distribution
Throw dart at a spinning wheel.
X
is the phase where the dart hits the wheel.
Find the cdf of
X
.
First we note the range of the random variable
X,
{0
2 }
X
Sx
π
=
≤≤
.
Within the range,
[
/ 2
/ 2]
/ 2 for any phase
.
Px
X x
x
−∆
≤
≤ +∆
=∆
Therefore,
[ ]
12
() P
0
2
2
00
X
x
x
Fx
X
x
x
x
≥
=
≤
=
≤
<
Homework:
Plot the cdf of
(1,3).
U
Example  Bernoulli
Let
X
be a Bernoulli random variable.
1
with prob
0
for a head
p
X
for a tail
=
,
11
( )
P[
]
1
0
1
00
X
x
Fx
X
x
p
x
x
≥
=
≤
=
−
≤<
<
Discrete RVs have discontinuities in their cdf’s.
The values of their cdf’s are taken approaching from the right.
1
1
1
p
−
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document Properties:
①
0
() 1
X
Fx
≤≤
②
()1
X
F
∞=
③
(
)0
X
F
−∞ =
④
()
X
is a nondecreasing function of
x
⑤
X
is continuous from the right: that is,
0
( )
lim
(
)
XX
h
Fb
h
→
=
probability density function ( pdf)
Definition:
0
(
)
()
() l
im
XX
Fx
d
f
x
dx
∆→
+∆ −
=
∆
Differentiate the cdf to get the pdf
A pdf is the rate at which its cdf increases
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview. Sign up
to
access the rest of the document.
This note was uploaded on 11/23/2010 for the course EE EE528 taught by Professor Majungsoo during the Spring '10 term at Korea Advanced Institute of Science and Technology.
 Spring '10
 Majungsoo

Click to edit the document details