- A joint probability distribution is used to describe a probability distribution over various variables. 
- If all possible states are equally likely, we describe it as a uniform distribution.
- The description will depend on whether the distribution consists of discrete or continuous variables.

Ohio State University, Columbus

Population distributions are generally used to describe the probability of a random variable or set of random variables taking on each of its possible states.

Probability Distribution

Goodfellow, I., Bengio, Y., & Courville, A. (2016). $\mathit{Deep \ Learning.}$ MIT Press. Retrieved from [www.deeplearningbook.org](https://www.deeplearningbook.org) 

Deep Learning

A PMF is used to describe a probability distribution.
- P(𝑥) is used to denote the probability that x = 𝑥
- A probability of 0 means x = 𝑥 is always false
- A probability of 1 means x = 𝑥 is always true

Probability Mass Function (PMF)

Probability Distribution Description

A data scientist is modeling the possible outcomes of a single event, which can result in one of six states (State 1 through State 6). Which of the following sets of probabilities represents a valid probability distribution for this event?

A machine learning model is designed to classify an animal image into one of three possible categories: 'Cat', 'Dog', or 'Bird'. For a specific image, the model outputs the following likelihoods for each category: P(Cat) = 0.5, P(Dog) = 0.6, P(Bird) = -0.1. Analyze this output and explain, with reference to the fundamental properties of a probability distribution, why this set of values is invalid.

Validating a Model's Probability Output

An AI model is designed to classify customer feedback into one of three mutually exclusive categories: 'Positive', 'Neutral', or 'Negative'. For a particular piece of feedback, the model outputs the following probabilities:

- P(Positive) = 0.65
- P(Neutral) = 0.25

Due to a software glitch, the probability for the 'Negative' category is missing. Based on the fundamental properties that govern the probabilities of all possible outcomes for a single event, what must be the probability for the 'Negative' category? Explain your reasoning.

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