12.3. Properties of PMF

1. Non-Negativity:

   $P(x) \geq 0$ for all $x$.

The first property of a Probability Mass Function (PMF) is non-negativity. This means that for every possible value of the discrete random variable, the probability assigned to that value must be zero or greater. In simpler terms, we cannot have negative probabilities, as they do not make sense in the context of probability. Every outcome must at least have a probability of zero, indicating that it is impossible, or a positive probability, indicating that it can occur.

Think of it like measuring the likelihood of different weather outcomes for a day. The chance of it raining or being sunny can’t be negative. If the chance of a sunny day is 0.7, it means there’s a 70% likelihood it will be sunny, and thus it cannot be less than zero.

2. Normalization (Total Probability = 1):

   $\sum P(x) = 1$.

The second property is normalization, which states that when you add up the probabilities of all possible outcomes of the random variable, the total must equal one. This reflects the certainty principle in probability: if you account for all possible outcomes, one of them must happen. Normalization ensures that the entire probability distribution is complete and coherent.

Imagine a game of rolling a fair die. The die has six faces, and the probabilities for each face showing up (1/6 for each number 1 to 6) must add up to 1. If you check these probabilities, you'll find that (1/6 + 1/6 + 1/6 + 1/6 + 1/6 + 1/6) = 1, confirming that you are certain one of those outcomes is going to happen when you roll the die.

3. Discrete Domain: $P(x)$ is defined only for countable values of $x$.

The final property of a PMF is that it is only defined for discrete (countable) values of the random variable. This means probabilities are assigned to specific outcomes that can be enumerated, such as the number of heads in a series of coin tosses or the result of rolling a die. This property distinguishes PMFs from Probability Density Functions (PDFs), which apply to continuous variables.

Think of it like counting apples. If you want to know how many different ways you can have 0, 1, 2, or 3 apples in a basket, you're dealing with a discrete situation since you can count those outcomes. In contrast, if you were measuring the weight of the apples, you would be dealing with a continuous range of possibilities rather than discrete values.

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