The following table collects properties of probability distribution functions. In our notation we reserve the label p(x) for the probability of a certain event, while P(x) is the cumulative probability.
| Discrete PDF | Continuous PDF | |
| Domain | \left\{x_1, x_2, x_3, \dots, x_N\right\} | [a,b] |
| Probability | p(x_i) | p(x)dx |
| Cumulative | P_i=\sum_{l=1}^ip(x_l) | P(x)=\int_a^xp(t)dt |
| Positivity | 0\le p(x_i)\le 1 | p(x) \ge 0 |
| Positivity | 0\le P_i\le 1 | 0\le P(x)\le 1 |
| Monotonic | P_i\ge P_j if x_i\ge x_j | P(x_i)\ge P(x_j) if x_i\ge x_j |
| Normalization | P_N=1 | P(b)=1 |