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A Probability Mass Function (PMF) is a critical aspect of understanding probability within discrete sample spaces. It allows us to assign a probability to every possible outcome of a discrete random variable, which can take on only specific distinct values. The PMF is defined mathematically as a function, where each value that the variable can take is matched with a probability that falls between 0 and 1.

• To be valid, the sum of all these probabilities for all possible outcomes in the sample space must equal one.
• Each individual probability must be non-negative, meaning it must be greater than or equal to zero.

In the given problem, the PMF is expressed by the function \( m(j) = (1-r)^j r \), where \( 0 < r < 1 \). With this formula, each outcome \( j \) in the sample space is assigned a probability. The function ensures that total probability is distributed across all possible outcomes such that they sum up to one, making it a valid PMF.

Discrete probability is a branch of probability that focuses on events that occur in specific, separate outcomes. It deals with scenarios where outcomes can be counted, and are typically finite or countably infinite. Because the outcomes are distinct, this kind of probability lays its foundation on a set of countable outcomes. The probability of each outcome is determined by the probability mass function. Within discrete probability, each event has a probability between 0 and 1, where a probability of 0 signifies an impossible outcome and a probability of 1 signifies certainty. In the exercise, we see discrete probabilities at play because the sample space consists of all non-negative integers: \( 0, 1, 2, \ldots \). The function \( m(j) = (1-r)^j r \) represents these discrete probabilities, providing a framework to calculate the probability of discrete outcomes occurring within the defined sample space.

The concept of a sample space is fundamental when discussing probability. It is the complete set of all possible outcomes of a probabilistic experiment. In our exercise, the sample space is given as \( \Omega = \{ 0, 1, 2, \ldots \} \), representing the set of non-negative integers. This sample space is infinite, highlighting an important aspect of the problem: the PMF must properly account for this potentially infinite set of outcomes. Each point in the sample space is associated with a probability determined by the PMF. The exercise defined \( m(j) = (1-r)^j r \), ensuring that the probabilities for outcomes in this space are valid and well-defined. By summing up the probabilities across the sample space, they need to align with the principle that total probability is one, a requirement for any proper probability distribution.Understanding the sample space helps in applying and interpreting the rules of probability, as it sets the stage on which probabilities play out.