Вопрос
What is a probability space? What is a set of measure zero and why can a finite or countable set have probability zero in a continuous distribution?
Ответить самому
Сначала сформулируйте ответ как на собеседовании, затем откройте разбор и оцените себя.
Короткий ответ
A probability space is (Omega, F, P): outcomes, measurable events and a probability measure. A measure-zero set has P(A)=0; under continuous distributions, individual points and countable unions of points can have zero probability.
Полный разбор
A probability space consists of three parts. Omega is the sample space of possible outcomes. F is a sigma-algebra of events we are allowed to assign probabilities to. P is a probability measure on F with P(Omega)=1, P(empty)=0 and countable additivity.
The sigma-algebra matters because for continuous spaces not every arbitrary subset is well-behaved under a measure. It is closed under complements and countable unions, which lets probability operations remain consistent.
A set of measure zero is an event with probability zero. In a continuous distribution over R, a single exact point has probability zero, and any countable set of points also has probability zero by countable additivity. This does not mean the event is logically impossible; it means it carries no probability mass under that measure.
Теория
Probability theory is measure theory with total mass one.
Типичные ошибки
- Say probability zero means impossible.
- Forget the sigma-algebra component.
- Use finite additivity when countable additivity is the key property.
Как отвечать на собеседовании
- State the triple (Omega, F, P) first.
- Use continuous uniform distribution on [0,1] as the simplest example.