Consider the function
fx=120fx=120
for
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, inclusive .

* fx=120fx=120
for
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