|
Size: 669
Comment: Initial commit
|
Size: 949
Comment: Moving to analysis
|
| Deletions are marked like this. | Additions are marked like this. |
| Line 1: | Line 1: |
| ## page was renamed from Statistics/ExpectedValues | |
| Line 3: | Line 4: |
| An '''expected value''' is the estimated outcome of an event. The math notation is ''E[A]''. | An '''expected value''' is the estimated outcome of an event. The math notation is ''E[X]''. |
| Line 13: | Line 14: |
| For a discrete distribution, the expected value is generally ''Σ x P(x)''. | For a discrete distribution, the expected value of ''X'' is generally ''Σ x P(x)'' (for all ''X=x''). |
| Line 15: | Line 16: |
| For a continuous distribution, the expected value is generally ''∫ x P(x) dx''. | For a continuous distribution, the expected value of ''X'' is generally ''∫ x P(x) dx'' (for all ''X=x''). |
| Line 31: | Line 32: |
| ---- == 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]