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In mathematics, a piecewise function is a function composed of multiple sub-functions, each of which applies to a specific interval within the function's domain. This kind of function can be very useful in various fields, including statistics and probability.
One common example of where piecewise functions are used is in defining a cumulative distribution function (CDF). When you look at CDFs, a piecewise function may tell us different probabilities over different ranges of a variable. For instance, in the exercise provided, the CDF has several intervals with distinct probability values. The structure of piecewise functions allows for the modeling of systems that exhibit different behavior over different ranges. Depending on the value of the variable, different rules apply, making it a powerful tool for probability descriptions.
In practice, reading a piecewise function involves carefully understanding which interval a certain value falls into and then applying the relevant sub-function. By doing this, one can accurately determine things like probabilities or other outcomes, dictated by these sub-functions.

Probability calculation involves determining the likelihood of certain outcomes within a specified range. When using a cumulative distribution function (CDF), this process typically involves identifying the correct portion of the function to apply based on the variable range.
For example, in the given exercise, the CDF can be used to calculate probabilities such as \( P(X \leq 50) \). This operation involves recognizing where the value of 50 falls within the piecewise intervals of the function and applying the corresponding probability value.
Key steps in probability calculations like these include:

• Identify which piece of the piecewise function applies to the value or range.
• Use the right part of the piecewise function to get the relevant probability.
• For ranges such as \( P(40 \leq X \leq 60) \), ensure to subtract the CDF values at the endpoints to find the probability.

This process helps in understanding how likely an event is according to the cumulative distribution provided.

A random variable is a variable whose possible values are numerical outcomes of a random phenomenon. In probability and statistics, random variables are used to quantify and understand random processes.
In the context of a cumulative distribution function (CDF), a random variable \( X \) represents the outcomes over which the CDF is defined. In the exercise, \( X \) could represent various real-life quantities, such as measurements or counts, that exhibit randomness. The CDF helps us to understand the probability that a particular random variable takes on a value less than or equal to a certain number.
Some important points to remember about random variables include:

• They can be discrete or continuous, with discrete variables having specific separated values and continuous ones having a range of values.
• The cumulative distribution function provides a complete description of the probability distribution for a random variable.
• When analyzing random variables, the CDF is a tool that helps quantify the likeliness of achieving certain values or being within a range.

Understanding random variables and their distributions can offer valuable insights into the behavior of systems or processes that incorporate elements of chance.