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The Cumulative Distribution Function (CDF) is a fundamental concept in probability theory that describes the probability that a random variable takes on a value less than or equal to a certain amount. When you have a random variable, say \( Y \), the CDF, represented as \( F(y) \), provides the probability that \( Y \) is less than or equal to \( y \), i.e., \( F(y) = P(Y \leq y) \).

This function comes handy for both discrete and continuous variables. It has several important properties:

• It is non-decreasing, which means it never decreases as \( y \) increases.
• It is right continuous, meaning it does not have jumps if approached from the right.
• It ranges between 0 and 1 inclusive.

In the context of our exercise, the CDF \( F(y) \) helps determine the discrete probability \( p(y) \) for different integer values.

A Discrete Random Variable is a type of random variable that takes on countable values, typically integers. In other words, the possible outcomes can be listed out individually, such as 1, 2, 3, and so on.

Discrete random variables are distinct from continuous random variables, which take on any value within a range. Here are some key points about discrete random variables:

• Each value of a discrete random variable can be associated with a specific probability.
• The sum of all probabilities of a discrete random variable is 1.
• These random variables are often used to model scenarios where the outcomes are finite and countable.

The exercise illustrates this concept using a random variable \( Y \) with possible integer values. By utilizing the cumulative distribution function (CDF), we compute probability values for each of these integer outcomes.

Piecewise Functions are mathematical expressions that are defined by different formulas or rules over different intervals of their domain. Such functions are important for modeling situations where a relationship or pattern changes at certain points.

In the context of probability, a piecewise function can describe probabilities that depend on different cases or scenarios, like the probabilities based on the discrete random variable and its CDF.

For example, in our task, the probability function \( p(y) \) is defined as a piecewise function given by:

• \( F(1) \) for \( y = 1 \)
• \( F(y) - F(y-1) \) for \( y = 2, 3, \ldots \)

This piecewise function is key to accurately representing the probability distribution of the random variable \( Y \), depending on whether \( y \) is 1 or some other integer value.