Linearity of Expectation
Expectation of sums is sum of expectations.
Key idea
Expectation of sums is sum of expectations.
In depth
E[X + Y] = E[X] + E[Y] always, even when X and Y are dependent. It makes complex counting problems easy: break the count into indicator variables and add their means.
Worked example
Expected number of fixed points in a random permutation of n items is n × (1/n) = 1.
Formula
Real-world uses
- Estimating the average number of matching birthdays or guests sharing a room without full counting
Common mistake
Assuming E[XY] = E[X]E[Y]; that requires independence.
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 Linearity of Expectation?
Expectation of sums is sum of expectations. It is a topic at the expert level in Probability & Statistics.
What is the formula for Linearity of Expectation?
The key formula is E[X+Y] = E[X]+E[Y].
Where is Linearity of Expectation used in real life?
Real-world examples: Estimating the average number of matching birthdays or guests sharing a room without full counting.
What should I learn before Linearity of Expectation?
Make sure you are comfortable with Variance of a Random Variable, the previous topic in Probability & Statistics.
What is a common mistake with Linearity of Expectation?
Assuming E[XY] = E[X]E[Y]; that requires independence.
How can I practise Linearity of Expectation?
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 Linearity of Expectation?
Move on to Bernoulli Distribution 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.