Lecture 20  Friday May 15th
[email protected]
Key words: Probability distribution function, Gaussian distribution
Key concepts: Know how to show that
f
(
x
) =
1
√
2
π
e

x
2
/
2
is a pdf.
20.1 Probability and Expectation
Recall a function
f
:
R
2
→
R
defines a
probability distribution function
on
R
n
if
f
(
x
)
≥
0
for all
x
and
Z Z
· · ·
Z
R
n
fdV
= 1
.
Underlying the probability distribution function is a probability measure defined by some
experiment which gives an outcome to each point in
R
n
.
If
X
is the outcome of the
experiment, then the probability distribution function gives
P
(
X
≥
λ
) =
Z Z
· · ·
Z
E
fdV
where
E
=
{
x
∈
R
n
:
X
≥
λ
}
– so
E
is the set of points in
R
n
which witness the outcome
of the experiment being at least
λ
. A basic example of a probability distribution function
is the
uniform distribution
. Here we define
f
(
x, y
) = 1 for (
x, y
)
∈
D
and
D
= [0
,
1]
×
[0
,
1]
and
f
(
x, y
) = 0 otherwise. Then
f
(
x, y
)
≥
0 for all (
x, y
)
∈
R
2
and
RR
D
fdA
= 1, so
f
is
indeed a probability distribution function. Now suppose the experiment we perform is to
measure the distance from a random point in
D
to the origin. Let
X
be the outcome of
this experiment, so that
X
(
x, y
) =
p
x
2
+
y
2
. Now consider the event
X
≥
1. The points
which witness this outcome are the points (
x, y
)
∈
D
such that
x
2
+
y
2
≥
1. Then we see
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.
 Spring '07
 Enright
 Math, Normal Distribution, Probability, Probability theory, probability density function, probability distribution function, Galton board

Click to edit the document details