# 2.2 Key functions and four models

Definition 2.1 *The cumulative distribution function, also called the distribution function and usually denoted F*

_{X}(

*x*)

*or F*(

*x*),

^{1}

*for a random variable X is the probability that X is less than or equal to a given number. That is, F*

_{X}(

*x*) =

*Pr*(

*X ≤ x*).

*The abbreviation*

**cdf**is often used.The distribution function must satisfy a number of requirements^{2}:

*F*(

*x*) ≤ 1 for all

*x.*

*F*(

*x*) is nondecreasing.

_{x→-∞}

*F*(

*x*) = 0 and lim

_{x→∞}

*F*(

*x*) = 1.

Because it need not be left-continuous, it is possible for the distribution function to jump. When it jumps, the value is assigned to the top of the jump.

Here are possible distribution functions for each of the four models.

**Model 1**^{4} This random variable could serve as a model for the age at death. All ages between 0 and 100 are possible. While experience suggests that there is an upper bound for human lifetime, models with no upper limit may be useful if they assign extremely low probabilities to extreme ages. This allows the modeler to avoid setting a specific maximum age:

This cdf is illustrated ...

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