8.1. The Uniform Probability Density:
Example:
X: the random variable representing the flight time from Taipei to Kaohsiung.
Suppose the flight time can be any value in the interval from 30 to 50 minutes. That is,
30  X  50. .
Question: if the probability of a flight time within any time interval
is the same as the one within the other time interval with the same
length. Then, what density f (x) is sensible for describing the
probability?
Recall that the area under the graph of f (x) corresponding to any interval is the
probability of the random variable X taking values in this interval. Since the
probabilities of X taking values in any equal length interval are the same, then the the
areas under the graph of f (x) corresponding to any equal length interval are the
same. Thus, f (x) will take the same value over any equal length area. For example,
within one minute interval, then
31
32
50
30
31
49
P(30  X  31)   f ( x)dx  P(31  X  32)   f ( x)dx    P(49  X  50)   f ( x)dx
Therefore, we have
f ( x) 
1
, 30  x  50; f ( x)  0, otherwise.
20
Note: since we know
f ( x)  c  some constant , then by the property
that
50
50
30
30
 f ( x)dx   cdx  1  20c  1  c 
1
1
20 .
In the above example, the probability density has the same value in the interval the
random variable taking value. This probability density is referred as the uniform
probability density function.
Uniform Probability Density Function:
A random variable X taking values in [a,b] has the uniform
probability density function f (x) if
f ( x) 
1
, a  x  b; f ( x)  0, otherwise.
ba
.
f(x)
The graph of f (x) is
1/(b-a)
a
b
x
Properties of Uniform Probability Density Function:
A random variable X taking values in [a,b] has the uniform
probability density function f (x) , then
ba
E( X ) 
,
2
2

b  a
Var ( X ) 
12
2
[Derivation]:
1
1 x2 b
1  b2 a2 
  
E ( X )   xf ( x)dx   x 
dx 
 |a 
ba
ba 2
ba 2 2 
a
a
b

b
1 b  a b  a  b  a


ba
2
2
The derivation of Var ( X ) 
b  a 2
12
is left as an exercise.
Example:
In the flight time example, b  50, a  30, then
50  30  33.33
50  30
E( X ) 
 40, Var ( X ) 
2
12
2
Online Exercise:
Exercise 8.1.1
Exercise 8.1.2
3