Lecture 11
AMS 311, Spring Semester, 2010
Chapter 5: Continuous Random Variables
5.1. Introduction
The random variable
X
is continuous if there exists a nonnegative function
X
f
, defined
for all real
x
having the property that for any set
B
of real numbers
.
)
(
}
{
∫
=
∈
B
X
dx
x
f
B
X
P
The function
X
f
is called the probability density function (pdf).
5.2. Expectation and Variance of Continuous Random Variables
Lemma 2.1
.
For any nonnegative random variable
Y
,
.
}
[
)
(
0
∫
∞
=
dy
y
Y
P
Y
E
Proposition 2.1
.
If
X
is a continuous random variable with probability density function
f(x)
; then for any
function
g
:
R
R
→
,
E
g
X
g
x
f
x
d x
(
(
) )
(
)
(
)
.
=
 ∞
∞
∫
5.3. The Uniform Random Variable
A random variable is said to be uniformly distributed over the interval (0,1) if its pdf is
given by
,
1
0
,
1
)
(
<
<
=
x
x
f
and zero otherwise. A random variable
X
is uniformly
distributed over the interval
)
,
(
β
α
if its pdf is
,
,
1
)
(
<
<

=
x
x
f
and zero
otherwise. Then
,
2
)
(
+
=
X
E
and
.
12
)
(
)
var(
2

=
X
5.4. Normal Random Variables
The random variable
X
is normally distributed with mean
μ
and variance
2
σ
if its
probability density function is
.
,
2
1
)
(
)
2
/(
)
(
2
2
∞
<
<
∞

=


x
e
x
f
x
X
π
A normal
random variable with mean 0 and variance 1 is called a standard normal random variable.
Its cdf is given in the equation
.
)
(
2
/
2
∫
∞


=
Φ
x
z
dz
e
x
This cdf is tabulated and is a basic
reference for working problems; see page 203 of your text. All probabilities are
calculated through conversion to a standard normal distribution.
The tables that I will
give you in the next examination and the final are below.
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View Full DocumentNormal Probability Table, AMS 311 Examinations
The table below (copied from the Actuarial series of examinations) give the value of
∫
∞


=
Φ
x
w
dw
e
x
2
/
2
2
1
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 Spring '08
 Tucker,A
 Normal Distribution, Probability theory, density function, continuous random variable, cdf FX

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