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 | \( P_N=1 \) | \( P(b)=1 \) |