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Probability theory serves as the foundation for understanding how likely different outcomes are, given some conditions. It helps us quantify uncertainty and make informed decisions based on the likelihood of events.
In the context of cumulative distribution functions (CDFs), probability theory provides us with the tools to describe the distribution of a random variable. A random variable could be anything measurable that can take on different values. For example, the roll of a dice, the height of people in a city, or the temperature on any given day.
Key principles of probability theory include:

• For any event, the probability ranges between 0 and 1, where 0 means the event will not occur, and 1 means the event will certainly occur.
• The sum of probabilities of all possible outcomes of a random event is 1.

Understanding these fundamentals allows us to explore more complex topics like nondecreasing functions and probability mass functions, which can describe how outcomes are distributed.

A function is called nondecreasing if, for any two points in its domain, if the first point is smaller than the second, the value of the function at the first point is less than or equal to its value at the second point.
For cumulative distribution functions (CDFs), this property ensures that as you move to the right along the x-axis (considering higher values of the variable), the probability either stays the same or increases.
This makes intuitive sense — if you're considering a larger value, you're including more possible outcomes, which means the total probability should not decrease. In mathematical terms, for a CDF \(F(x)\), we have:

• \(x_1 < x_2\) leads to \(F(x_1) \leq F(x_2)\)

The only time a CDF would remain flat (constant probability) over an interval is when there's no event occurring with values in that interval, meaning the probability mass within that interval is zero.
This reflects the condition in the problem where \(F(x_1) = F(x_2)\), highlighting intervals devoid of probability mass.

A Probability Mass Function (PMF) is used to describe the probability of outcomes for discrete random variables. Each possible discrete value the random variable can take has a probability mass associated with it, representing how likely it is to occur.
PMFs are essential for understanding cumulative distribution functions (CDFs) because they inform how probability accumulates over a distribution. For a CDF, each increment in its value corresponds to an event that contributes a probability mass from the PMF.

• A PMF assigns probabilities such that their total sums to 1, covering all possible values.
• Each probability mass is non-negative, meaning each discrete outcome has a chance of occurring, or it is impossible, but not less than zero.

In our example, \(P(x_1 < X \leq x_2) = 0\) describes a condition in which the probability mass is concentrated outside the interval from \(x_1\) to \(x_2\) or the specific event happens with probability zero in that interval. Thus, the CDF in nondecreasing functions doesn’t increase, showing their reliance on probability mass functions to define the wider probabilistic structure.