|
Size: 684
Comment: Initial commit
|
← Revision 9 as of 2025-11-23 02:16:57 ⇥
Size: 1355
Comment: Expected values
|
| Deletions are marked like this. | Additions are marked like this. |
| Line 3: | Line 3: |
| A joint probability is the likelihood of multiple events occurring, either simultaneously or sequentially. The joint probability of two events is notated as ''P(A ∩ B)'', or sometimes more informally as ''P(AB)''. | A '''joint probability''' is the likelihood of multiple events occurring. |
| Line 11: | Line 11: |
| == Decomposition == | == Description == |
| Line 13: | Line 13: |
| The intersection of two events ''A'' and ''B'' is the same as event ''A'' times the [[Statistics/ConditionalProbability|conditional probability]] of ''B'' given ''A''. | This is the probability of a set of two (or more) events occurring. This coincidence of events is an '''intersection''', which notated for e.g. two events as ''A∩B''. |
| Line 15: | Line 15: |
| This can be expanded out to any number of events. | In some cases, especially when the events feature binary outcomes, an intersection is informally notated like ''P(AB)''. |
| Line 17: | Line 17: |
| ''P(A,,1,, ∩ A,,2,, ∩ ... ∩ A,,n-1,, ∩ A,,n,,) = P(A,,1,,|A,,2,, ∩ ... ∩ A,,n,,) ... P(A,,n-1,,|A,,n,,) P(A,,n,,) | These events can be sequential or simultaneous. To demonstrate the validity of the latter, note that the probability of ''A∩B'' is the same as the probability of ''B'' multiplied by the [[Statistics/ConditionalProbability|conditional probability]] of ''A'' given ''B'': ''P(A∩B) = P(A|B) P(B)''. === Expected Values === For discretely distributed X and Y, the [[Statistics/ExpectedValues|expected value]] is generally ''Σ,,y,,Σ,,x,, x y P(x,y)''. For continuously distributed X and Y, the expected value is generally ''∫∫,,Ω,, x y P(x,y) dxdy''. === Independence === '''Independence''' means the occurrence of one event has no impact on the probability of another event. For example, given two independent events ''A'' and ''B'', the joint probability is simply the product of their individual probabilities: ''P(A∩B) = P(A) P(B)''. |
Joint Probability
A joint probability is the likelihood of multiple events occurring.
Description
This is the probability of a set of two (or more) events occurring. This coincidence of events is an intersection, which notated for e.g. two events as A∩B.
In some cases, especially when the events feature binary outcomes, an intersection is informally notated like P(AB).
These events can be sequential or simultaneous. To demonstrate the validity of the latter, note that the probability of A∩B is the same as the probability of B multiplied by the conditional probability of A given B: P(A∩B) = P(A|B) P(B).
Expected Values
For discretely distributed X and Y, the expected value is generally ΣyΣx x y P(x,y).
For continuously distributed X and Y, the expected value is generally ∫∫Ω x y P(x,y) dxdy.
Independence
Independence means the occurrence of one event has no impact on the probability of another event. For example, given two independent events A and B, the joint probability is simply the product of their individual probabilities: P(A∩B) = P(A) P(B).