a function describes how probabilities are distributed over the values of the random variable.
Example A (continue):
Suppose the probability for the outcomes in the gamble example is
P ({ Win })

1
6

P ( X

3 )

P ( Y

30 )
P ({ Lose })

2
3

P ( X
P ({ Tie })

1
6

P ( X
 
4 )

P ( Y

0 )

P ( Y

0 )
 
40 )
Let f x
( x ) be some function corresponding to the probability of the gambling outcomes for random variable X , defined as f f x
( 3 )

P ( X f x
(

4 )

P ( X x
( 0 )

P ( X

3 )

1
6
 
4 )

2
3

0 )

1
6 f x
( x ) is referred as the probability distribution of random variable X.
Similarly, the probability distribution f y
( x ) of random variable Y is f f y
( 30 )

P ( Y f y
(

40 )

P ( Y

30 )

1
6
 
40 )

2
3 y
( 0 )

P ( Y

0 )

1
6
1

Let a
1
, a
2
,  , a n
,  be all the possible values of the discrete random variable X. Then, the required conditions for f ( x ) to be the discrete probability distribution for X are
(a) f ( a i
)

0 , for every i.
(b) i
 f ( a i
)
 f ( a
1
)
 f ( a
2
)
   f ( a n
)
  
1
Example A (continue):
In the gambling example, f x
( x ) is a discrete probability distribution for the random
(a) variable X since f x
( 3 )

0 , f x
(

4 )

0 , and f x
( 0 )

0
.
(b) f x
( 3 )
 f x
(

4 )
 f x
( 0 )

1
.
Similarly, f y
( x ) is also a discrete probability distribution for the random variable Y .
the discrete probability distribution describes the probability of a discrete random variable at different values.
For a continuous random variable, it is impossible to assign a probability to every numerical value since there are uncountable number of values in an interval. Instead, the probability can be assigned to a small interval. The probability density function can describe how the probability distributes in the small interval.
Example B (continue):
In the delay flight time example, suppose the probability of being late within 0.5 hours is two times of the one of being late more than 0.5 hour, i.e.,
P ( 0

Z

0 .
5 )

2
3 and P ( 0 .
5

Z

1 )

1
3
.
Then, the probability density function f
1
( x ) for the random variable Z is
2

f
1
( x )

4
3
, 0
 x

0 .
5 ; f
1
( x )

2
3
, 0 .
5
 x

1 .
0.0
0.2
0.4
0.6
0.8
1.0
x
The area corresponding to the interval is the probability of the random variable Z taking values in this interval. For example, the probability of the flight time being late within 0.5 hour (the random variable Z taking value in the interval [0,0.5]). is
P ( The flight tim e being late within 0.5
hour)

P ( 0

Z

0 .
5 )

0

0 .
5 f
1
( x ) dx

4
3
* 0 .
5

2
3
.
Similarly, the probability of the flight time being late more than 0.5 hour (the random variable Z taking value in the interval (0.5,1]). is
P ( The flight tim e being late more than 0.5
hour)

P ( 0 .
5

Z

1 )

1

0 .
5 f
1
( x ) dx

2
3
* 0 .
5

1
3
.
On the other hand, If the probability of being late within 0.5 hours is the same as the one of being late more than 0.5 hour, i.e.,
P ( 0

Z

0 .
5 )

P ( 0 .
5

Z

1 )

1
2
, then, the probability density function f
2
( x )
 f
2
( x ) for the random variable Z is
1 , 0
 x

1 .
Note that the probability density function corresponds to the probability of the random variable taking values in some interval. However, the probability density function evaluated at some value, not like the probability distribution,
3

can not be used to describe the probability of the random variable Z taking this value.
Let the continuous random variable Z taking values in [a,b]. Then, the required conditions for f ( x ) to be the continuous probability density for Z are
(a) f ( x )

0 , a
 x
 b .
(b) b a
 f ( x ) dx

1
P ( c

Z
 d )
  d c f ( x ) dx , a
 c
 d
 b
. That is, the area under the graph of f ( x ) corresponding to a given interval is the probability of the random variable Z taking value in this interval.
Example B (continue):
In the flight time example, f
1
( x ) is a continuous probability density for the random variable Z since
(a) f
1
( x )

0 , 0
 x

1
.
(b)
0 .
5

0
4
3 dx
 
1
0 .
5
2
3 dx

1
.
Similarly, f
2
( x ) is also a continuous probability density for the random variable Z .
Exercise 6.2.1
4