We begin by defining a continuous probability density function. We use the function notation
fXfX. Intermediate algebra may have been your first formal introduction to functions. In the
study of probability, the functions we study are special. We define the function fXfX so that the
area between it and the x-axis is equal to a probability. Since the maximum probability is one,
the maximum area is also one.
For continuous probability distributions, PROBABILITY = AREA.
Consider the function
fX=120fX=120
where
0<X<20
0
X
20.
XX = a real number.
The graph of
fX=120fX=120 is a horizontal line. However, since
0<X<20
0
X
20
,
fXfX is restricted to
the portion between X=0X0 and
X=20X20 .
fX=120fX=120
where
0<X<20
0
X
20.
The graph of fX=120fX=120
is
a horizontal line segment
when
0<X<20
0
X
20.
The area between fX=120fX=120
where
0<X<20
0
X
20
and the x-axis is the area of a rectangle
with base = 2020 and height =120120.
AREA=20⋅120=1
AREA
20⋅120
1
This particular function, where we have restricted XX so that the area
between the function and the x-axis is 1, is an example of a continuous
probability density function. It is used as a tool to calculate
probabilities.
Suppose we want to find the area between fX=120fX=120
and the x-axis where 0<X<2
0
X
2
.
AREA=(2-0)⋅
120=0.1
AREA
(2-0)⋅
120
0.1
(2-0)=2= base of a rectangle
(2-0)
2
base of a rectangle
120120 = the height.
The area corresponds to a probability. The probability that XX is between 0 and 2 is 0.1, which can be written mathematically as P(0<X<2) = P(X<2)=0.1P(0<X<2)= P(X<2)=0.1.
Suppose we want to find the area between fX=120fX=120
and the x-axis where 4<X<15
4
X
15
.
AREA=(15-4)⋅
120=0.55
AREA
(15-4)⋅
120
0.55
(15-4) = 11 = the base of a rectangle(15-4)=11=the base of a rectangle
120120 = the height.
The area corresponds to the probability
P(4<X<15)=0.55
P(4
X
15)
0.55.
Suppose we want to find P(X = 15)P(X = 15). On an x-y graph, X = 15X = 15 is a vertical line. A vertical
line has no width (or 0 width). Therefore, P(X = 15) = (base)(height) = (0)(120) = 0P(X = 15)=(base)(height)= (0)(120)=0.
P(X≤x)
P(X
x)
(can be written as
P(X<x)
P(X
x)
for continuous distributions) is called the
cumulative distribution function or CDFCDF. Notice the "less than or equal to" symbol. We
can use the CDFCDF to calculate
P(X>x)
P(X
x)
. The CDFCDF gives "area to the left" and
P(X>x)
P(X
x)
gives "area to the right." We calculate
P(X>x)
P(X
x)
for continuous distributions as follows:
P(X>x)=1-P(X<x)
P(X
x)
1-P(X
x).
Label the graph with
f(X)f(X) and XX.
Scale the x and y axes with the maximum xx and yy values.
fX=120
fX
120,
0<X<20
0
X
20.
P(2.3<X<12.7)=(base) (height)=(12.7-2.3)
(120)=0.52
P(2.3
X
12.7)
(base) (height)
(12.7-2.3)
(120)
0.52
The previous problem is an example of the uniform probability distribution.
"This book was purchased from the authors by the Maxfield Foundation and provided to the community as an open textbook available freely online and in PDF format. Bound copies of the book can also […]"