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If the events are not mutually exclusive then
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[[Category:Formulas]]
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[[Category:probability]]
  
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    <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>
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<center><font size= 4>
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'''[[Collective_Table_of_Formulas|Collective Table of Formulas]]'''
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</font size>
  
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Conditional probability is written ''P''(''A''|''B''), and is read "the probability of ''A'', given ''B''"
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Probability Formulas
  
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    <math>P(A \mid B) = \frac{P(A \cap B)}{P(B)}\,</math>
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click [[Collective_Table_of_Formulas|here]] for [[Collective_Table_of_Formulas|more formulas]]
  
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</center>
  
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----
 
{|
 
{|
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! colspan="2" style="background:  #e4bc7e; font-size: 110%;" | Discrete-time Fourier Transform Pairs and Properties
 
 
|-
 
|-
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! colspan="2" style="background: #eee;" | Property of Probability Functions
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! style="background: rgb(228, 188, 126) none repeat scroll 0% 0%; -moz-background-clip: -moz-initial; -moz-background-origin: -moz-initial; -moz-background-inline-policy: -moz-initial; font-size: 110%;" colspan="2" | Probability Formulas
 
|-
 
|-
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| align="right" style="padding-right: 1em;" | The complement of an event A (i.e. the event A not occurring) || <math>\,P(A^c) = 1 - P(A)\,</math>
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! style="background: rgb(238, 238, 238) none repeat scroll 0% 0%; -moz-background-clip: -moz-initial; -moz-background-origin: -moz-initial; -moz-background-inline-policy: -moz-initial;" colspan="2" | Properties of Probability Functions
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|-  
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| align="right" style="padding-right: 1em;" | The intersection of two independent events A and B || <math>\,P(A \mbox{ and }B) =  P(A \cap B) = P(A) P(B)\,</math>
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|-
 
|-
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| align="right" style="padding-right: 1em;" | The union of two events A and B (i.e. either A or B occurring) || <math>\,P(A \mbox{ or } B) = P(A) + P(B) - P(A \mbox{ and } B)\,</math>
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| align="right" style="padding-right: 1em;" | The complement of an event A (i.e. the event A not occurring)  
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| <math>\,P(A^c) = 1 - P(A)\,</math>
 
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|-
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| 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>
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| align="right" style="padding-right: 1em;" | The intersection of two independent events A and B  
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|}
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| <math>\,P(A \mbox{ and }B) =  P(A \cap B) = P(A) P(B)\,</math>
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{|
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|-
 
|-
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! colspan="4" style="background: #eee;" | DT Fourier Transform Pairs
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| align="right" style="padding-right: 1em;" | The union of two events A and B (i.e. either A or B occurring)  
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|-
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| <math>\,P(A \mbox{ or } B) = P(A) + P(B) - P(A \mbox{ and } B)\,</math>
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| align="right" style="padding-right: 1em;" |   || <math>x[n]</math> || <math>\longrightarrow</math>|| <math> \mathcal{X}(\omega) </math>
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| 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>
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|-
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|-
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| align="right" style="padding-right: 1em;" | The union of two mutually exclusive events A and B
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| align="right" style="padding-right: 1em;" | || <math>a^{n} u[n], |a|<1 \ </math> || ||<math>\frac{1}{1-ae^{-j\omega}} \ </math>
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| <math>\,P(A \mbox{ or } B) =  P(A \cup B)= P(A) + P(B)\,</math>
 
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|-
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| 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>
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| align="right" style="padding-right: 1em;" | Event A occurs given that event B has occurred
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| <math>\,P(A \mid B) = \frac{P(A \cap B)}{P(B)}\,</math>
 
|-
 
|-
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|}
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| align="right" style="padding-right: 1em;" | Total Probability Law
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| <math>\,P(B) = P(B|A_1)P(A_1) + \dots + P(B|A_n)P(A_n)\,</math>
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<math> \mbox{ where } \{A_1,\dots,A_n\} \mbox{ is a partition of sample space } S, B \mbox{ is an event }.</math>
  
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{|
 
 
|-
 
|-
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! colspan="4" style="background: #eee;" | DT Fourier Transform Properties
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| align="right" style="padding-right: 1em;" | Bayes Theorem
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|-
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| <math>\,P(A_j|B) = \frac{P(B|A_j)P(A_j)}{\sum_{i=1}^{n}P(B|A_i)P(A_i)},\ \{A_i\} \mbox{ and } B \mbox{ are as above }.</math>
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| align="right" style="padding-right: 1em;" |   || <math>x[n]</math> || <math>\longrightarrow</math>|| <math> \mathcal{X}(\omega) </math>
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|-
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| 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>
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|-
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| align="right" style="padding-right: 1em;" |  convolution property || <math>x[n]*y[n] \!</math> || ||<math> X(\omega)Y(\omega) \!</math>
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|-
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| align="right" style="padding-right: 1em;" | time reversal ||<math>\ x[-n] </math> || ||<math>\ X(-\omega)</math>
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! style="background: rgb(238, 238, 238) none repeat scroll 0% 0%; -moz-background-clip: -moz-initial; -moz-background-origin: -moz-initial; -moz-background-inline-policy: -moz-initial;" colspan="2" |  Expectation and Variance of Random Variables
 
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|-
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|}
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| align="right" style="padding-right: 1em;" | Binomial random variable with parameters n and p
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| <math>\,E[X] = np,\ \ Var(X) = np(1-p)\,</math>
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{|
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|-
 
|-
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! colspan="2" style="background: #eee;" | Other DT Fourier Transform Properties
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| align="right" style="padding-right: 1em;" | Poisson random variable with parameter <math>\lambda</math>
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|-  
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| <math>\,E[X] = \lambda,\ \ Var(X) = \lambda\,</math>
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| align="right" style="padding-right: 1em;" | Parseval's relation  || <math>\frac {1}{N} \sum_{n=-\infty}^{\infty}\left| x[n] \right|^2 = </math>
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|-
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| align="right" style="padding-right: 1em;" | Geometric random variable with parameter p
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| <math>\,E[X] = \frac{1}{p},\ \ Var(X) = \frac{1-p}{p^2}\,</math>
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|-
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| align="right" style="padding-right: 1em;" | Uniform random variable over (a,b)
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| <math>\,E[X] = \frac{a+b}{2},\ \ Var(X) = \frac{(b-a)^2}{12}\,</math>
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|-
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| align="right" style="padding-right: 1em;" | Gaussian random variable with parameter <math>\mu \mbox{ and } \sigma^2</math>
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| <math>\,E[X] = \mu,\ \ Var(X) = \sigma^2\,</math>
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|-
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| align="right" style="padding-right: 1em;" | Exponential random variable with parameter <math>\lambda</math>
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| <math>\,E[X] = \frac{1}{\lambda},\ \ Var(X) = \frac{1}{\lambda^2}\,</math>  
 
|}
 
|}
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[[Collective_Table_of_Formulas|Back to Collective Table]]  
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==Relevant Courses==
 +
*[[ECE600|ECE600]]
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*[[ECE302|ECE302]]
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----
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[[Collective Table of Formulas|Back to Collective Table]]  
 +
 
 
[[Category:Formulas]]
 
[[Category:Formulas]]

Latest revision as of 11:54, 3 March 2015


Collective Table of Formulas

Probability Formulas

click here for more formulas

Probability Formulas
Properties 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)}\, $
Total Probability Law $ \,P(B) = P(B|A_1)P(A_1) + \dots + P(B|A_n)P(A_n)\, $

$ \mbox{ where } \{A_1,\dots,A_n\} \mbox{ is a partition of sample space } S, B \mbox{ is an event }. $

Bayes Theorem $ \,P(A_j|B) = \frac{P(B|A_j)P(A_j)}{\sum_{i=1}^{n}P(B|A_i)P(A_i)},\ \{A_i\} \mbox{ and } B \mbox{ are as above }. $
Expectation and Variance of Random Variables
Binomial random variable with parameters n and p $ \,E[X] = np,\ \ Var(X) = np(1-p)\, $
Poisson random variable with parameter $ \lambda $ $ \,E[X] = \lambda,\ \ Var(X) = \lambda\, $
Geometric random variable with parameter p $ \,E[X] = \frac{1}{p},\ \ Var(X) = \frac{1-p}{p^2}\, $
Uniform random variable over (a,b) $ \,E[X] = \frac{a+b}{2},\ \ Var(X) = \frac{(b-a)^2}{12}\, $
Gaussian random variable with parameter $ \mu \mbox{ and } \sigma^2 $ $ \,E[X] = \mu,\ \ Var(X) = \sigma^2\, $
Exponential random variable with parameter $ \lambda $ $ \,E[X] = \frac{1}{\lambda},\ \ Var(X) = \frac{1}{\lambda^2}\, $

Relevant Courses

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