Probability Theory
Probability and statistics are fields concerned with mathematically describing and analyzing uncertain phenomena and uncovering patterns from data. WIIS provides systematic learning materials on modern probability theory and statistics grounded in measure theory, with a focus on probability spaces, random variables, discrete probability distributions, continuous probability distributions, and asymptotic theory.
Field of Probability and Statistics
This section introduces the axiomatic approach to probability theory. After defining concepts such as probability spaces and probability functions, we derive and prove the fundamental properties of probability spaces from the axioms of probability.
New Materials
When a sequence of countably many events is monotone increasing or monotone decreasing, a property known as continuity holds for the probabilities of their union or intersection.
Symmetric difference events are measurable. The probability of a symmetric difference event is obtained by subtracting twice the probability of the intersection from the sum of the probabilities of the two events.
Difference events are measurable. The probability of the difference event \(A\backslash B\) is obtained by subtracting the probability of the intersection event \(A\cap B\) from the probability of \(A\).
Intersection events are measurable. If the probabilities of two events and the probability of their union are known, the probability of their intersection can be derived from the addition theorem. If the probability of the union is unknown, Bonferroni's inequality can be used to narrow down the possible range of the probability of the intersection.
By the axioms of probability, the complement of any event is measurable. Moreover, the probability of the complement of an event \(A\) is obtained by subtracting the probability of \(A\) from \(1\), which is the probability of the entire sample space. We derive this result from the axioms of probability.
Union events are measurable. The additivity of the probability measure is useful for determining the probability of the union of mutually disjoint events, while subadditivity and the addition theorem are useful for unions of events that are not necessarily mutually disjoint.
Learning Guide
Related Fields
Set theory is the common language of modern mathematics. Focusing on sets, functions, relations, cardinality, and order relations, this course systematically explains the fundamental concepts needed to understand a wide range of mathematical fields. It provides a foundation for further study in areas such as real analysis, linear algebra, and topology.