Discrete Random Variables & Probability Distributions
Discrete random variables take separate values, each with a probability; the probabilities of a valid distribution add to \(1\).
Part of the NSW Year 12 Mathematics Advanced course, in the Statistical analysis area of study (Random variables focus area) of the 2024 syllabus. Work through practice questions with fully worked solutions and video lessons, or scroll down for the theory summary and worked examples.
Theory
A discrete random variable takes separate values, each with a probability that all add to 1. This Year 12 Mathematics Advanced topic (MAV-12-07) works with probability distributions.
\(P(X=x)\) is the probability \(X\) takes the value \(x\); each \(0\le P(x)\le 1\) and \(\sum P(x)=1\). A uniform distribution over \(n\) values has each probability \(\dfrac1n\).
Method
- Use \(\sum P(x)=1\) to find a missing probability.
- Add probabilities for cumulative events like \(P(X\le k)\).
- Uniform: each value has probability \(\dfrac1n\).
| \(P(3)\) | \(=\) | \(1-0.6=0.4\) |
| \(P(X\le2)\) | \(=\) | \(0.2+0.3=0.5\) |
| \(P(X=4)\) | \(=\) | \(\dfrac16\) |
| \(10k\) | \(=\) | \(1\) |
| \(k\) | \(=\) | \(0.1\) |
Common pitfalls
Frequently asked questions
What is a discrete random variable?
A variable that takes separate values, each with a probability, such as the number rolled on a die.
What makes a valid probability distribution?
Every probability is between 0 and 1, and they all add up to 1.
What is a uniform distribution?
One where every value is equally likely, so each has probability 1 over n.
How do you find a missing probability?
Use the fact that all the probabilities add to 1.