Difference between revisions of "Probability Formulas" - Rhea

Revision as of 07:57, 22 October 2010


Property of Probability Functions
The complement of an event A (i.e. the event A not occurring)	$\,P(A^c) = 1 - P(A)\,$
The intersection of two independent events A and B	$\,P(A \mbox{ and }B) = P(A \cap B) = P(A) P(B)\,$
The union of two events A and B (i.e. either A or B occurring)	$\,P(A \mbox{ or } B) = P(A) + P(B) - P(A \mbox{ and } B)\,$
The union of two mutually exclusive events A and B	$\,P(A \mbox{ or } B) = P(A \cup B)= P(A) + P(B)\,$
Event A occurs given that event B has occurred	$\,P(A \mid B) = \frac{P(A \cap B)}{P(B)}\,$

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-If the events are not mutually exclusive then
-    <math>\mathrm{P}\left(A \hbox{ or } B\right)=\mathrm{P}\left(A\right)+\mathrm{P}\left(B\right)-\mathrm{P}\left(A \mbox{ and } B\right)</math>
-Conditional probability is written ''P''(''A''|''B''), and is read "the probability of ''A'', given ''B''"
-    <math>P(A \mid B) = \frac{P(A \cap B)}{P(B)}\,</math>
 {|
-! colspan="2" style="background:  #e4bc7e; font-size: 110%;" | Discrete-time Fourier Transform Pairs and Properties
+! colspan="2" style="background:  #e4bc7e; font-size: 110%;" |
 |-
 ! colspan="2" style="background: #eee;" | Property of Probability Functions
@@ Line 20: / Line 11: @@
 |-
 | align="right" style="padding-right: 1em;" | The union of two mutually exclusive events A and B || <math>\,P(A \mbox{ or } B) =  P(A \cup B)= P(A) + P(B)\,</math>
-|}
-{|
 |-
-! colspan="4" style="background: #eee;" | DT Fourier Transform Pairs
+| align="right" style="padding-right: 1em;" | Event A occurs given that event B has occurred || <math>\,P(A \mid B) = \frac{P(A \cap B)}{P(B)}\,</math>
-|-
-| align="right" style="padding-right: 1em;" |   || <math>x[n]</math> || <math>\longrightarrow</math>|| <math> \mathcal{X}(\omega) </math>
-|-
-| align="right" style="padding-right: 1em;" | DTFT of a complex exponential || <math>e^{jw_0n}</math> || || <math>\pi\sum_{l=-\infty}^{+\infty}\delta(w-w_0-2\pi l) \ </math>
-|-
-|-
-| align="right" style="padding-right: 1em;" |  || <math>a^{n} u[n],  |a|<1 \ </math> || ||<math>\frac{1}{1-ae^{-j\omega}} \ </math>
-|-
-| align="right" style="padding-right: 1em;" |  || <math>\sin\left(\omega _0 n\right) u[n] \ </math>  || ||<math>\frac{1}{2j}\left( \frac{1}{1-e^{-j(\omega -\omega _0)}}-\frac{1}{1-e^{-j(\omega +\omega _0)}}\right)</math>
-|-
-|}
-{|
-|-
-! colspan="4" style="background: #eee;" | DT Fourier Transform Properties
-|-
-| align="right" style="padding-right: 1em;" |   || <math>x[n]</math> || <math>\longrightarrow</math>|| <math> \mathcal{X}(\omega) </math>
-|-
-| align="right" style="padding-right: 1em;" | multiplication property|| <math>x[n]y[n] \ </math> || || <math>\frac{1}{2\pi} \int_{2\pi} X(\theta)Y(\omega-\theta)d\theta</math>
-|-
-| align="right" style="padding-right: 1em;" |  convolution property || <math>x[n]*y[n] \!</math> || ||<math> X(\omega)Y(\omega) \!</math>
-|-
-| align="right" style="padding-right: 1em;" | time reversal ||<math>\ x[-n] </math> || ||<math>\ X(-\omega)</math>
-|-
-|}
-{|
-|-
-! colspan="2" style="background: #eee;" | Other DT Fourier Transform Properties
-|-
-| align="right" style="padding-right: 1em;" | Parseval's relation  || <math>\frac {1}{N} \sum_{n=-\infty}^{\infty}\left| x[n] \right|^2 = </math>
 |}
 ----
 [[Collective_Table_of_Formulas|Back to Collective Table]]
 [[Category:Formulas]]

Difference between revisions of "Probability Formulas" - Rhea

Revision as of 07:57, 22 October 2010

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