Cumulative Distribution Functions
Definition
The cumulative distribution function F (x) for a continuous rv X is
defined for every number x by
Z x
F (x) = P(X ≤ x) =
f (y )dy
−∞
For each x, F (x) is the area under the density curve to the left of x.
Liang Zhang (UofU)
Applied Statistics I
June 26, 2008
1 / 11
Cumulative Distribution Functions
Example 4.6
Let X , the thickness of a certain metal sheet, have a uniform distribution
on [A, B]. The pdf for X is
(
1
A≤x ≤B
f (x) = B−A
0
otherwise
Then the cdf for X is calculated as following:
For x < A, F (x) = 0; for A ≤ x < B, we have
Z x
Z x
1
x −A
1
dy =
· y |yy =x
F (x) =
f (y )dy =
=A = B − A ;
B
−
A
B
−
A
−∞
A
for x ≥ B, F (x) = 1.
Therefore the entire cdf for X is


0
x−A
F (x) = B−A


1
Liang Zhang (UofU)
x <A
A≤x <B
x ≥B
Applied Statistics I
June 26, 2008
2 / 11
Cumulative Distribution Functions
Proposition
Let X be a continuous rv with pdf f (x) and cdf F (x). Then for any
number a,
P(X > a) = 1 − F (a)
and for any two numbers a and b with a < b,
P(a ≤ X ≤ b) = F (b) − F (a).
Liang Zhang (UofU)
Applied Statistics I
June 26, 2008
3 / 11
Cumulative Distribution Functions
Example (Problem 15)
Let X denote the amount of space occupied by an article placed in a 1-ft3
packing container. The pdf of X is
(
90x 8 (1 − x) 0 < x < 1
f (x) =
0
otherwise
Then what is P(X ≤ 0.5) and P(0.25 < X ≤ 0.5)?
Liang Zhang (UofU)
Applied Statistics I
June 26, 2008
4 / 11
Cumulative Distribution Functions
Proposition
If X is a continuous rv with pdf f (x) and cdf F (x), then at every x at
which the derivative F 0 (x) exists, F 0 (x) = f (x).
e.g. for the previous example, we know the


0
F (x) = 10x 9 − 9x 10


1
cdf for X is
x ≤0
0<x <1
x ≥1
Then the derivative of F (x) exists on (−∞, ∞) and we get
F 0 (x) = 90x 8 − 90x 9 for 0 < x < 1 and F 0 (x) = 0 for −∞ < x ≤ 0 and
1 ≤ x < ∞, which is just the pdf of X .
Liang Zhang (UofU)
Applied Statistics I
June 26, 2008
5 / 11
Cumulative Distribution Functions
Definition
The expected value or mean valued of a continuous rv X with pdf f (x)
is
Z ∞
µX = E (X ) =
x · f (x)dx
−∞
Definition
The variance of a continuous random variable X with pdf f (x) and mean
value µ is
Z ∞
2
σX = V (X ) =
(x − µ)2 · f (x)dx = E [(X − µ)2 ]
−∞
The standard deviation (SD) of X is σX =
Liang Zhang (UofU)
Applied Statistics I
p
V (X ).
June 26, 2008
6 / 11
Cumulative Distribution Functions
Proposition
V (X ) = E (X 2 ) − [E (X )]2
e.g. for the previous example, the pdf of X is given as
(
90x 8 (1 − x) 0 < x < 1
f (x) =
0
otherwise
Then the expected value of X is
Z ∞
Z
E (X ) =
x · f (x)dx =
−∞
Z
= 90
0
Liang Zhang (UofU)
1
x · 90x 8 (1 − x)dx
0
1
(x 9 − x 10 )dx = 90(
1 10
1
9
x − x 11 ) |x=1
x=0 =
10
11
11
Applied Statistics I
June 26, 2008
7 / 11
Cumulative Distribution Functions
Example continued: the pdf for X is
(
90x 8 (1 − x) 0 < x < 1
f (x) =
0
otherwise
The variance of X is
V (X ) = E (X 2 ) − [E (X )]2 =
Z
∞
x 2 · f (x)dx − [
−∞
Z
∞
x · f (x)dx]2
−∞
Z 1
=
x 2 · 90x 8 (1 − x)dx − [
x · 90x 8 (1 − x)dx]2
0
0
Z 1
9
= 90
(x 10 − x 11 )dx − ( )2
11
0
1 11
1 12 x=1
9
= 90( x − x ) |x=0 −( )2
11
12
11
15
81
3
=
−
=
22 121
242
Z
1
Liang Zhang (UofU)
Applied Statistics I
June 26, 2008
8 / 11
Cumulative Distribution Functions
Definition
Let p be a number between 0 and 1. The (100p)th percentile of the
distribution of a continuous rv X , denoted by η(p), is defined by
Z
η(p)
p = F (η(p)) =
f (y )dy
−∞
In words, the (100p)th percentile η(p) is the X value such that there are
100p% X values below η(p).
Graphically, η(p) is the value on the measurement axis such that 100p% of
the area under the graph of f (x) lies to the left of η(p) and 100(1 − p)%
lies to the right.
Liang Zhang (UofU)
Applied Statistics I
June 26, 2008
9 / 11
Cumulative Distribution Functions
Liang Zhang (UofU)
Applied Statistics I
June 26, 2008
10 / 11
Cumulative Distribution Functions
Definition
The median of a continuous distribution, denoted by µ̃, is the 50th
percentile, so µ̃ satisfies 0.5 = F (µ̃). That is, half the area under the
density curve is to the left of µ̃ and half is to the right of µ̃.
e.g. for the continuous rv X with cdf


0
F (x) = 10x 9 − 9x 10


1
x ≤0
0<x <1
x ≥1
the 100pth percentile is calculated as following:
p = F (η(p)) = 10η(p)9 − 9η(p)10
Therefore, the 75th percentile is η(.75) ≈ 0.9036 and the median is
η(.5) ≈ 0.8377.
Liang Zhang (UofU)
Applied Statistics I
June 26, 2008
11 / 11