Understanding Probability Distributions

Probability distributions are ways to show how likely different outcomes are for two types of random variables: discrete and continuous.

Discrete Random Variables
Discrete random variables have specific, separate values. This means they can only be certain numbers.

For example, think about rolling a fair six-sided die. Each side of the die shows a different number: 1, 2, 3, 4, 5, or 6.

The chance of rolling any one of these numbers is the same:

• Each has a probability of ( \frac{1}{6} ).

This means all possible outcomes add up to 1. You can clearly see that each outcome is separate and different.

Continuous Random Variables
On the other hand, continuous random variables can take on any number within a range.

Their probabilities are explained using something called a probability density function (PDF).

For example, let’s say we measure the height of students. The heights can range from 150 cm to 200 cm.

Instead of giving a specific chance for each possible height, we look at ranges.

For any exact height ( x ), the chance ( P(X = x) ) is actually 0.

To find the probability for a certain height range, like between 160 cm and 170 cm, we use tools from calculus.

Specifically, we calculate the probability as:

• ( P(a < X < b) = \int_a^b f(x) , dx ).

Wrapping It Up
In short, discrete distributions deal with clear, separate chances for each outcome.

Continuous distributions, however, focus on the chance of outcomes falling within certain ranges.