|
Size: 723
Comment: Notation
|
← Revision 3 as of 2024-03-21 20:11:42 ⇥
Size: 897
Comment: Linearity
|
| Deletions are marked like this. | Additions are marked like this. |
| Line 31: | Line 31: |
| ---- == Linearity == Expectations are linear. ''E[X + c] = E[X] + c'' ''E[X + Y] = E[X] + E[Y]'' ''E[aX] = a E[X]'' ''E[aX + bY] = a E[X] + b E[Y]'' |
Expected Values
An expected value is the estimated outcome of an event. The math notation is E[X].
Contents
Evaluation
For a discrete distribution, the expected value of X is generally Σ x P(x) (for all X=x).
For a continuous distribution, the expected value of X is generally ∫ x P(x) dx (for all X=x).
Bernoulli
For a Bernoulli-distributed variable (taking value 1 with probability p and value 0 with probability 1-p), the expected value is p.
E[X] = Σ x P(x)
E[X] = (0) P(0) + (1) P(1)
E[X] = P(1)
E[X] = p
Linearity
Expectations are linear.
E[X + c] = E[X] + c
E[X + Y] = E[X] + E[Y]
E[aX] = a E[X]
E[aX + bY] = a E[X] + b E[Y]