5,337
edits
Probability: Difference between revisions
no edit summary
No edit summary |
No edit summary |
||
| Line 1: | Line 1: | ||
Calculus-based Probability | Calculus-based Probability | ||
==Axioms of Probability== | ==Basics== | ||
===Axioms of Probability=== | |||
* <math>0 \leq P(E) \leq 1</math> | * <math>0 \leq P(E) \leq 1</math> | ||
* <math>P(S) = 1</math> where <math>S</math> is your sample space | * <math>P(S) = 1</math> where <math>S</math> is your sample space | ||
| Line 11: | Line 12: | ||
==Expectation and Variance== | ==Expectation and Variance== | ||
Some definitions and properties. | Some definitions and properties. | ||
===Definitions=== | |||
Let <math>X \sim D</math> for some distribution <math>D</math>. | |||
Let <math>S</math> be the support or domain of your distribution. | |||
* <math>E(X) = \sum_S xp(x)</math> or <math>\int_S xp(x)dx</math> | |||
* <math>Var(X) = E[(X-E(X))^2] = E(X^2) - (E(X))^2</math> | |||
===Total Expection=== | ===Total Expection=== | ||
Dr. Xu refers to this as the smooth property. | Dr. Xu refers to this as the smooth property. | ||
<math>E(X) = E(E(X|Y))</math> | <math>E(X) = E(E(X|Y))</math> | ||
{{hidden | Proof |}} | {{hidden | Proof | | ||
<math> | |||
E(X) = \int_S xp(x)dx | |||
= \int_x x \int_y p(x,y)dy dx | |||
= \int_x x \int_y p(x|y)p(y)dy dx | |||
= \int_y\int_x x p(x|y)dxp(y)dy | |||
</math> | |||
}} | |||
===Total Variance=== | ===Total Variance=== | ||
This one is not used as often on tests as total expectation | This one is not used as often on tests as total expectation | ||