cumulative distribution function
/ CDF /
Often you do not care about the chance of an exact value but the chance of being at or below some level: what is the probability a claim is no larger than $10,000? The cumulative distribution function, or CDF, answers exactly that. For any number x it gives F(x) = P(X ≤ x), the running total of probability accumulated from the bottom up to x. It is probability's odometer.
The CDF always starts at 0 far to the left (nothing has accumulated yet), climbs to 1 far to the right (all probability accounted for), and never goes down — it only rises or stays flat. For a discrete variable it climbs in steps at each possible value; for a continuous variable it rises smoothly. From the CDF you can read off any interval probability by subtraction: P(a < X ≤ b) = F(b) - F(a). Quantiles run the CDF backward — the 95th percentile is the value x where F(x) = 0.95, the loss level you exceed only 5% of the time.
The CDF is the actuary's everyday lookup table. Percentiles drive the portfolio-percentile premium and the Value-at-Risk capital measure (a high quantile of the loss CDF). The survival function used throughout life contingencies is simply 1 - F(x), the chance of lasting beyond x. Whenever you ask 'what is the chance the loss stays under the limit?' or 'what level covers 99% of cases?', you are reading the CDF.
If F(10000) = 0.97 for a claim-size variable, then 97% of claims are $10,000 or less, and only 3% exceed it. The 97th percentile of the loss is $10,000.
The CDF reads off 'how much probability is at or below this level'.
The CDF can never decrease; if a calculation produces a 'probability of being below' that falls as the threshold rises, the model has an error.