We can construct the cumulative distribution for arandom variable where a random variable may only take certain discrete values - from the probability dens function.
Suppose the probability mass function for a random variable 
is
|
|
0 |
1 |
2 |
3 |
|
|
0.3 |
0.2 |
0.1 |
0.4 |
For the cumulative distribution functionm we want the values of
We can obtain these by adding up the individual probabilities for values of up to and including 
We obtain
|
|
0 |
1 |
2 |
3 |
|
|
0.3 |
0.3+0.2=0.5 |
0.3+0.2+0.1=0.6 |
0.3+0.2+0.1+0.4=1 |
Conversely, given a cumulative distribution function we can construct the probability mass function using the formula
From the cumulative distribution function above
|
|
0 |
1 |
2 |
3 |
|
|
0.3 |
0.5-0.3=0.2 |
0.6-0.5=0.1 |
1-0.6=0.4 |
This returns the original probability mass function.