Definition and Overview of Discrete Probability Distributions

A discrete probability distribution or probability mass function defines the probability of each possible outcome when the outcomes are discrete (countable) numbers.
The events described by a discrete probability distribution must be mutually exclusive and exhaustive. This means that one and only one outcome will occur on each trial, and that all possible outcomes must be accounted for.
Each individual outcome in a discrete probability distribution has a positive probability of happening. The sum of all these individual probabilities must equal to 1.

Key Characteristics

With discrete probability distributions, you can calculate the probability of specific outcomes. For example, if we have a discrete probability distribution representing the number of red marbles pulled from a bag, we can determine the probability of pulling exactly two red marbles.
Discrete probability distributions can be displayed in tables, graphs, or formulas. Tables provide a straightforward, easy-to-read view of outcomes and probabilities, while graphs and formulas provide a visual representation or calculation tool.
Common examples of discrete probability distributions include the Binomial distribution, Poisson distribution, and Geometric distribution.

To analyse a discrete probability distribution, statisticians often calculate a few key parameters or measures.
These include the expected value (mean), which represents the long-term average outcome, the variance, which indicates how spread out the outcomes are, and the standard deviation, which is the square root of the variance.
The expected value is calculated as E(X)= ∑ [x * P(x)], where ‘x’ is each outcome and ‘P(x)’ is the probability of each outcome.
The variance and standard deviation provide insights into the volatility or unpredictability of the outcomes. If the variance or standard deviation is large, the outcomes are widely spread around the mean.

Discrete probability distributions are extremely useful for modelling real-world situations where the outcomes are countable. They are used in fields as diverse as mathematics, physics, engineering, statistics, computer science, and economics.
For instance, they can describe the distribution of customers arriving at a store, the number of calls arriving at a call centre, or the number of emails you receive in a day.
Understand that the power of discrete probability distributions lies in their ability to provide sharp insights into the behaviour of random variables and allow for robust decision-making and prediction under uncertainty.

A crucial consideration is that not all probability distributions can be accurately represented with a discrete probability distribution. For instance, if the outcomes are not countable or the data is continuous, a continuous probability distribution is necessary.
The assumptions tied to each specific type of discrete probability distribution should always be checked before they can be applied. For example, a Poisson distribution assumes that events occur independently, and at a constant rate.
Interpret findings from discrete probability distributions with caution. These only provide mathematical probabilities, and real-world outcomes may not always follow these exact distributions due to unpredictable factors.