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A discrete random variable is a type of random variable that can take on a finite or countably infinite number of possible outcomes. In other words, it can only assume distinct and separate values. These values usually come from a countable set. For example, the roll of a die results in a discrete random variable, with possible outcomes of 1 through 6.

The key characteristic of discrete random variables is that you can list all the possible values they can take. This makes them different from continuous random variables, which can assume any value within a given range or interval.

The problem we are dealing with features a discrete random variable, denoted by \(X\). This random variable assumes values at specific points where the cumulative distribution function (CDF) jumps. These values represent the points where the probability mass is concentrated. Understanding this helps us analyze how likely certain outcomes are, which is crucial for calculating probabilities with these types of variables.

The cumulative distribution function (CDF) of a random variable tells us the probability that the variable takes on a value less than or equal to a certain level. Specifically, for a discrete random variable \(X\), the CDF \(F(a)\) gives the cumulative probability up to value \(a\). This function is often piecewise, with different rules or expressions for different intervals of \(a\).

In our example, the CDF is given for a discrete random variable \(X\):

• \(F(a) = 0\) for \(a < 0\)
• \(F(a) = \frac{1}{3}\) for \(0 \leq a < \frac{1}{2}\)
• \(F(a) = \frac{1}{2}\) for \(\frac{1}{2} \leq a < \frac{3}{4}\)
• \(F(a) = 1\) for \(a \geq \frac{3}{4}\)

These steps indicate the cumulative nature of the function: moving from smaller to larger \(a\) captures more probability. Each jump in the CDF occurs at a point where the PMF, or probability mass function, holds non-zero values. By examining these jumps, we can extract important information about the main discrete values of \(X\) and their associated probabilities.

Calculating probabilities for discrete random variables often involves analyzing its probability mass function (PMF). The PMF gives the probability that a discrete random variable takes exactly a specific value. This is important because these values define how probability is distributed across different possible outcomes for \(X\).

To determine the PMF from a given CDF, we look at the differences between cumulative probabilities across intervals. This is because, in a CDF, the increase between intervals represents the allocation of probability mass at specific discrete points.

For our example, from the CDF:

• The probability for \(X = 0\) is \(\frac{1}{3} - 0 = \frac{1}{3}\).
• For \(X = \frac{1}{2}\), it's \(\frac{1}{2} - \frac{1}{3} = \frac{1}{6}\).
• And for \(X = \frac{3}{4}\), it's \(1 - \frac{1}{2} = \frac{1}{2}\).

Thus, the PMF can be written as:

• \(p(x) = \frac{1}{3}\) if \(x = 0\)
• \(p(x) = \frac{1}{6}\) if \(x = \frac{1}{2}\)
• \(p(x) = \frac{1}{2}\) if \(x = \frac{3}{4}\)
• \(p(x) = 0\) otherwise

By following these calculations, you learn how the discrete random variable's behavior is quantified, allowing you to predict outcomes with defined probabilities.