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Probability: Difference between revisions
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* <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 | ||
* For mutually exclusive events <math>E_1, E_2, ...</math>, <math>P\left(\bigcup_i^\infty E_i\right) = \sum_i^\infty P(E_i)</math> | * For mutually exclusive events <math>E_1, E_2, ...</math>, <math>P\left(\bigcup_i^\infty E_i\right) = \sum_i^\infty P(E_i)</math> | ||
===Monotonicity=== | ===Monotonicity=== | ||
* For all events <math>A</math> and <math>B</math>, <math>A \subset B \implies P(A) \leq P(B)</math> | * For all events <math>A</math> and <math>B</math>, <math>A \subset B \implies P(A) \leq P(B)</math> | ||
{{hidden | Proof | }} | {{hidden | Proof | }} | ||
===PMF, PDF, CDF=== | |||
For discrete distributions, we call <math>p_{X}(x)=P(X=x)</math> the probability mass function (PMF).<br> | |||
For continuous distributions, we have the probability density function (PDF) <math>f(x)</math>.<br> | |||
The comulative distribution function (CDF) is <math>F(x) = P(X \leq x)</math>.<br> | |||
The CDF is the prefix sum of the PMF or the integral of the PDF. Likewise, the PDF is the derivative of the CDF. | |||
==Expectation and Variance== | ==Expectation and Variance== | ||