Probability vector

<h2 id="examples">Examples</h2>
<p>Here are some examples of probability vectors. The vectors can be either columns or rows.
</p>
<ul><li>
  
    
      
        
          x
          
            0
          
        
        =
        
          
            [
            
              
                
                  0.5
                
              
              
                
                  0.25
                
              
              
                
                  0.25
                
              
            
            ]
          
        
        ,
      
    
    {\displaystyle x_{0}={\begin{bmatrix}0.5\\0.25\\0.25\end{bmatrix}},}
  
</li>
<li>
  
    
      
        
          x
          
            1
          
        
        =
        
          
            [
            
              
                
                  0
                
              
              
                
                  1
                
              
              
                
                  0
                
              
            
            ]
          
        
        ,
      
    
    {\displaystyle x_{1}={\begin{bmatrix}0\\1\\0\end{bmatrix}},}
  
</li>
<li>
  
    
      
        
          x
          
            2
          
        
        =
        
          
            [
            
              
                
                  0.65
                
                
                  0.35
                
              
            
            ]
          
        
        ,
      
    
    {\displaystyle x_{2}={\begin{bmatrix}0.65&0.35\end{bmatrix}},}
  
</li>
<li>
  
    
      
        
          x
          
            3
          
        
        =
        
          
            [
            
              
                
                  0.3
                
                
                  0.5
                
                
                  0.07
                
                
                  0.1
                
                
                  0.03
                
              
            
            ]
          
        
        .
      
    
    {\displaystyle x_{3}={\begin{bmatrix}0.3&0.5&0.07&0.1&0.03\end{bmatrix}}.}
  
</li></ul>
<h2 id="geometric-interpretation">Geometric interpretation</h2>
<p>Writing out the vector components of a vector 
  
    
      
        p
      
    
    {\displaystyle p}
  
 as
</p>

p
        =
        
          
            [
            
              
                
                  
                    p
                    
                      1
                    
                  
                
              
              
                
                  
                    p
                    
                      2
                    
                  
                
              
              
                
                  ⋮
                
              
              
                
                  
                    p
                    
                      n
                    
                  
                
              
            
            ]
          
        
        
        
          or
        
        
        p
        =
        
          
            [
            
              
                
                  
                    p
                    
                      1
                    
                  
                
                
                  
                    p
                    
                      2
                    
                  
                
                
                  ⋯
                
                
                  
                    p
                    
                      n
                    
                  
                
              
            
            ]
          
        
      
    
    {\displaystyle p={\begin{bmatrix}p_{1}\\p_{2}\\\vdots \\p_{n}\end{bmatrix}}\quad {\text{or}}\quad p={\begin{bmatrix}p_{1}&p_{2}&\cdots &p_{n}\end{bmatrix}}}

<p>the vector components must sum to one:
</p>

∑
          
            i
            =
            1
          
          
            n
          
        
        
          p
          
            i
          
        
        =
        1
      
    
    {\displaystyle \sum _{i=1}^{n}p_{i}=1}

<p>Each individual component must have a probability between zero and one:
</p>

0
        ≤
        
          p
          
            i
          
        
        ≤
        1
      
    
    {\displaystyle 0\leq p_{i}\leq 1}

<p>for all 
  
    
      
        i
      
    
    {\displaystyle i}
  
. Therefore, the set of stochastic vectors coincides with the <a href="/facts/Simplex/0FCfNRN8">standard 
  
    
      
        (
        n
        −
        1
        )
      
    
    {\displaystyle (n-1)}
  
-simplex</a>. It is a point if 
  
    
      
        n
        =
        1
      
    
    {\displaystyle n=1}
  
, a segment if 
  
    
      
        n
        =
        2
      
    
    {\displaystyle n=2}
  
, a (filled) triangle if 
  
    
      
        n
        =
        3
      
    
    {\displaystyle n=3}
  
, a (filled) <a href="/facts/Tetrahedron/7mkSrl6j">tetrahedron</a> if 
  
    
      
        n
        =
        4
      
    
    {\displaystyle n=4}
  
, etc.
</p>
<h2 id="properties">Properties</h2>
<ul><li>The mean of the components of any probability vector is 
  
    
      
        1
        
          /
        
        n
      
    
    {\displaystyle 1/n}
  
.</li>
<li>The shortest probability vector has the value 
  
    
      
        1
        
          /
        
        n
      
    
    {\displaystyle 1/n}
  
 as each component of the vector, and has a length of 
  
    
      
        1
        
          /
        
        
          
            n
          
        
      
    
    {\textstyle 1/{\sqrt {n}}}
  
.</li>
<li>The longest probability vector has the value 1 in a single component and 0 in all others, and has a length of 1.</li>
<li>The shortest vector corresponds to maximum uncertainty, the longest to maximum certainty.</li>
<li>The length of a probability vector is equal to 
  
    
      
        
          
            n
            
              σ
              
                2
              
            
            +
            1
            
              /
            
            n
          
        
      
    
    {\textstyle {\sqrt {n\sigma ^{2}+1/n}}}
  
; where 
  
    
      
        
          σ
          
            2
          
        
      
    
    {\displaystyle \sigma ^{2}}
  
 is the variance of the elements of the probability vector.</li></ul>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Stochastic_matrix/2cRWAqC3">Stochastic matrix</a></li>
<li><a href="/facts/Dirichlet_distribution/Qq13f5g2">Dirichlet distribution</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Jacobs, Konrad (1992), Discrete Stochastics, Basler Lehrbücher [Basel Textbooks], vol. 3, Birkhäuser Verlag, Basel, p. 45, doi:10.1007/978-3-0348-8645-1, ISBN 3-7643-2591-7, MR 1139766. <a href="3-7643-2591-7" target="_blank">3-7643-2591-7</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
</ol>

Probability vector open-in-new

Probability vector