Independence

Compact study note.

Summary

Independence means that learning one event occurred does not change the probability of another. For random variables, independence is one statement about all joint events generated by those variables.^[1]

Prerequisites

• Conditional Probability

Notation and Assumptions

Event independence:

Random variable independence means, for all Borel sets $C,D$ ,

Essential Result

If $P(B)>0$ , event independence is equivalent to

Small Example

For two fair coin flips, 'first flip is heads' and 'second flip is heads' are independent because

Common Mistakes

• Treating disjoint events like independent; nonempty disjoint events are usually dependent.
• Checking only one pair of values and concluding random variables are independent.

Connections

• Conditional Probability
• IID Random Variables
• Covariance and Correlation

References