Conditional expectation

<p>In <a href="/facts/Probability_theory/mFBn51rE">probability theory</a>, the conditional expectation, conditional expected value, or conditional mean of a <a href="/facts/Random_variable/TwTBXnLT">random variable</a> is its <a href="/facts/Expected_value/1XV0JKL8">expected value</a> evaluated with respect to the <a href="/facts/Conditional_probability_distribution/0eGm3P9W">conditional probability distribution</a>. If the random variable can take on only a finite number of values, the "conditions" are that the variable can only take on a subset of those values. More formally, in the case when the random variable is defined over a discrete <a href="/facts/Probability_space/dEcv2VrU">probability space</a>, the "conditions" are a <a href="/facts/Partition_of_a_set/VeP9g8CA">partition</a> of this probability space.
</p><p>Depending on the context, the conditional expectation can be either a random variable or a function. The random variable is denoted 
  
    
      
        E
        (
        X
        ∣
        Y
        )
      
    
    {\displaystyle E(X\mid Y)}
  
 analogously to <a href="/facts/Conditional_probability/QcN2UERV">conditional probability</a>. The function form is either denoted 
  
    
      
        E
        (
        X
        ∣
        Y
        =
        y
        )
      
    
    {\displaystyle E(X\mid Y=y)}
  
 or a separate function symbol such as 
  
    
      
        f
        (
        y
        )
      
    
    {\displaystyle f(y)}
  
 is introduced with the meaning 
  
    
      
        E
        (
        X
        ∣
        Y
        )
        =
        f
        (
        Y
        )
      
    
    {\displaystyle E(X\mid Y)=f(Y)}
  
.
</p>

Conditional expectation open-in-new

Conditional expectation