School

Louisiana State University **We aren't endorsed by this school

Course

STAT 310

Subject

Statistics

Date

Aug 31, 2023

Pages

5

Uploaded by MinisterRiver11682 on coursehero.com

McGuffey
Spring 2023
Stat 310 Notes
L
ESSON
3.1:
C
ONTINUOUS
RV
S
: CDF, PDF, M
EAN
& V
ARIANCE
1.
Cumulative Distribution Functions
A cumulative distribution function (CDF) is another way to describe the distribution of a random
variable. Its definition is the same for discrete and continuous random variables.
C
UMULATIVE
D
ISTRIBUTION
F
UNCTIONS
The
cumulative distribution function
(CDF) of an r.v.
X
is the function
F
X
given by
Example 1.
Consider
X
with PMF
x
2
4
6
8
P
(
X
=
x
)
0.4
0.2
0.1
0.3
.
(a) Plot the PMF of
X
.
(b) Describe the CDF of
X
with a table.
That is, find
P
(
X
≤
x
) for
x
= 2, 4, 6, 8.
This is an
incomplete way to describe the CDF, but it helps us see what's going on.
(c) Plot the CDF of
X
.
Lesson 3.1
Page 1/5

McGuffey
Spring 2023
Stat 310 Notes
CDF Visualizations For Named Distributions
• Discrete and continuous:
https://www.geogebra.org/m/cQeyMaaO
!
4
Not our version of the Geometric distribution.
Example 2.
Consider a new r.v.
X
, with CDF plotted below.
Provide the PMF of
X
in table form.
CDF Properties
A valid CDF
F
X
has the following properties:
1.
: If
x
1
≤
x
2
, then
F
(
x
1
)
≤
F
(
x
2
).
2. Tends to
as
x
→ -∞
.
3. Tends to
as
x
→ ∞
.
2.
Continuous Random Variables
Formal definition: A random variable
X
is called
continuous
if there is a nonnegative function
f
X
called the
probability density function
(PDF) of
X
, such that
P
(
X
∈
B
) =
Z
B
f
X
(
x
)
dx
,
for every subset
B
of the real line.
Lesson 3.1
Page 2/5

McGuffey
Spring 2023
Stat 310 Notes
Discrete RVs
Continuous RVs
Possible
Values
distinct values
Distributions
PMF:
p
X
(
x
) =
P
(
X
=
x
)
PDF:
f
X
(
x
)
CDF:
F
X
(
x
) =
P
(
X
≤
x
)
CDF:
F
X
(
x
) =
P
(
X
≤
x
)
Valid PMF/PDF
•
Non-negative.
p
X
(
x
)
≥
0 for any
x
∈
R
•
Total mass of 1.
∑
x
p
X
(
x
) = 1
•
Non-negative.
f
X
(
x
)
≥
0 for any
x
∈
R
•
Total area of 1.
Discrete RVs
Continuous RVs
Probability
Calculations
P
(
a
≤
X
≤
b
) =
P
(
a
≤
X
≤
b
) =
X
a
≤
x
≤
b
p
X
(
x
)
P
(
X
=
b
) =
p
X
(
b
)
P
(
X
=
b
) =
P
(
X
≤
b
) =
∑
x
≤
b
p
X
(
x
)
P
(
X
≤
b
) =
=
F
X
(
b
)
Including
Endpoints
!
4
It matters.
On problem sets (but not exams), you may use
WolframAlpha
to evaluate integrals and take derivatives. On
exams, you may use an approved calculator. Always, you
must
include in your solution a statement of the
integral or derivative being evaluated.
Lesson 3.1
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