Cumulative Distribution Function
Probability that X ≤ x.
Key idea
Probability that X ≤ x.
In depth
F(x) = P(X ≤ x) accumulates probability up to x. It is non-decreasing, goes from 0 to 1, and for continuous variables F′(x) = f(x). Interval probabilities are P(a < X ≤ b) = F(b) − F(a).
Worked example
For a fair die, F(3) = 3/6 = 0.5.
Formula
Real-world uses
- Finding the chance a delivery arrives within 30 minutes or a sensor stays below a limit
Common mistake
Using the CDF where the probability of exactly equal (mass) is needed.
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 Cumulative Distribution Function?
Probability that X ≤ x. It is a topic at the expert level in Probability & Statistics.
What is the formula for Cumulative Distribution Function?
The key formula is F(x) = P(X ≤ x).
Where is Cumulative Distribution Function used in real life?
Real-world examples: Finding the chance a delivery arrives within 30 minutes or a sensor stays below a limit.
What should I learn before Cumulative Distribution Function?
Make sure you are comfortable with Continuous Random Variables, the previous topic in Probability & Statistics.
What is a common mistake with Cumulative Distribution Function?
Using the CDF where the probability of exactly equal (mass) is needed.
How can I practise Cumulative Distribution Function?
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 Cumulative Distribution Function?
Move on to Joint and Marginal Distributions 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.