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A discrete probability distribution describes the probability of occurrence of each value of a discrete random variable. Discrete variables can take on a finite or countably infinite set of values. In many cases, these values are integers. To visualize it, imagine a bag of marbles, each with a different number painted on it. Each marble corresponds to a value that the discrete random variable can take. The probability of each possible outcome is given by its probability mass function (PMF). The PMF uniquely defines the characteristics of the distribution, telling us the likelihood that a specific value will occur. Since the variable is discrete, the total probabilities of all possible outcomes must sum to 1. Discrete distributions are used in various situations, such as deciding how often an event happens in a fixed number of trials. Common discrete distributions include the Binomial, Poisson, and Geometric distributions.

In order to verify whether a function is a valid probability mass function (PMF), these two main criteria must be met:

• All probabilities must be greater than or equal to 0, meaning they are non-negative.
• The sum of the probabilities for all possible values must equal 1. This ensures that the entire probability space is accounted for.

Using the example function from the exercise, \( f(x) = \frac{8}{7}(\frac{1}{2})^x \), we calculated \( f(1) = \frac{4}{7} \), \( f(2) = \frac{2}{7} \), and \( f(3) = \frac{1}{7} \). Summing these values, we see that the total is \( 1 \).This verification conforms to the properties of discrete probability functions, indicating that the function is a valid PMF. It’s essential to check these conditions whenever identifying or using a probability mass function.

Calculating probabilities in a discrete distribution involves determining the specific probabilities of interest based on the probability mass function (PMF). Let's explore how we perform these calculations using the PMF provided.

• **For \( P(X \leq 1) \)**: This calculates the probability that \( X \) is less than or equal to 1, which results directly from \( f(1) = \frac{4}{7} \).
• **For \( P(X > 1) \)**: Here, \( X \) can be 2 or 3. We sum these probabilities: \( f(2) + f(3) = \frac{2}{7} + \frac{1}{7} = \frac{3}{7} \).
• **For \( P(2 < X < 6) \)**: Only \( x = 3 \) fulfills the condition, resulting in \( f(3) = \frac{1}{7} \).
• **For \( P(X \leq 1 \text{ or } X > 1) \)**: This probability aggregates all possible outcomes \( f(1) + f(2) + f(3) = 1 \), following the sum discussed in our verification step.

Calculating such probabilities helps us understand the likelihood of different outcomes within a discrete distribution, enabling clear insights into the randomness of the events being studied.