Continuous Random Variables
Use density functions; probability is area.
Key idea
Use density functions; probability is area.
In depth
A continuous variable takes any value in an interval. Probabilities are areas under a density curve f(x), where P(a ≤ X ≤ b) = ∫ f. The probability at any single point is 0.
Worked example
X uniform on [0, 1] has density 1; P(0.2 ≤ X ≤ 0.5) = 0.3.
Formula
Real-world uses
- Modelling a waiting time or a length measured to any precision in engineering
Common mistake
Reading the density as a probability; density can exceed 1.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Continuous Random Variables?
Use density functions; probability is area. It is a topic at the expert level in Probability & Statistics.
What is the formula for Continuous Random Variables?
The key formula is P(a<X<b) = ∫f(x)dx.
Where is Continuous Random Variables used in real life?
Real-world examples: Modelling a waiting time or a length measured to any precision in engineering.
What should I learn before Continuous Random Variables?
Make sure you are comfortable with Exponential Distribution, the previous topic in Probability & Statistics.
What is a common mistake with Continuous Random Variables?
Reading the density as a probability; density can exceed 1.
How can I practise Continuous Random Variables?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Continuous Random Variables?
Move on to Cumulative Distribution Function next.
Related expert topics in Probability & Statistics
- Skewness and Kurtosis – Measures of asymmetry and tail heaviness.
- Residuals and R² – Error of a fit and proportion of variance explained.
- Simpson’s Paradox – A trend reverses when groups are combined.
- Law of Total Probability – Split an event over a partition.
- Bayes’ Theorem – Update beliefs with evidence.
- Medical Tests and Base Rates – Why positive results can still be probably false.