Probability Distributions

Definition

The probability distribution function informs us what the probabilities of a range of outcomes defined in a random variable will be.

$$R_{x}=[1,2,3,4]$$

$$P_{x}=[0.95, 0.02, 0.01, 0.02]$$

Here, $R_{x}$ is the random variable containing some outcomes, and the values in $P_{x}$ represent the probabilities of those outcomes, respectively.

Probabilities can be displayed of simple occurrences like tossing a coin to more complex occurrences, like the probability of a specific steroid treating a disease or not.

Data types

There are mainly two types of data that we encounter while figuring out the probability of events:

• Discrete data: A random variable X has discrete data if the data it contains is finite, countable, and has specific values. For example, the outcome of a number of students in a class can only be a specific number like 50 or 60 and cannot be 50.5 or 57.7. If a die is rolled, the only outcomes are 1, 2, 3, 4, 5, and 6, which are finite, countable, and specific whole numbers.

• Continuous data: A random variable X has continuous data if all the data points it contains are in specific ranges. The range can either be finite or infinite. For example, if we have the data for heights of different buildings, we cannot find the probability that the height of a building is exactly 700 cm, but we can find the probability that the height of a building is between 650 - 750 cm. The height can be 700.075 or 699.874, but not exactly 700.

Types of the probability distribution

There are different types of probability distribution functions used to model various kinds of data. For this course, we will discuss the most commonly used probability distributions.

Uniform distribution

Events, where each and every outcome in a random variable has the same probability of occurrence will create a uniform distribution of probabilities. This means that if a random variable has n number of outcomes, each of those outcomes has a probability of occurrence equal to 1/n.

Uniform probability distribution can be created for both discrete and continuous data.

• Discrete uniform distribution: It uses discrete data and contains a finite number of outcomes that all have the same probability of occurrence. For example, a rolling die and a coin toss are excellent examples of this. The probability of all the outcomes of both of the above cases is the same, and since they contain a finite number of outcomes, they form a discrete uniform distribution.

• Continuous uniform distribution: It uses continuous data, and the outcomes are either in a range or infinite and have the same probability of occurrence. For example, a random number generator is an excellent example of this. In this example, there is an endless number of outcomes that can exist, and every number has an equal possibility of occurrence. So, a continuous uniform distribution is formed for this.