kb/data/en.wikipedia.org/wiki/Conditional_variance-1.md

title	chunk	source	category	tags	date_saved	instance
Conditional variance	2/2	https://en.wikipedia.org/wiki/Conditional_variance	reference	science, encyclopedia	2026-05-05T12:22:02.721176+00:00	kb-cron

=== Definition using conditional distributions === The "conditional expectation of Y given X=x" can also be defined more generally using the conditional distribution of Y given X (this exists in this case, as both here X and Y are real-valued). In particular, letting

      P
      
        Y
        
          |
        
        X
      
    
  

{\displaystyle P_{Y|X}}

be the (regular) conditional distribution

      P
      
        Y
        
          |
        
        X
      
    
  

{\displaystyle P_{Y|X}}

of Y given X, i.e.,

      P
      
        Y
        
          |
        
        X
      
    
    :
    
      
        B
      
    
    ×
    
      R
    
    →
    [
    0
    ,
    1
    ]
  

{\displaystyle P_{Y|X}:{\mathcal {B}}\times \mathbb {R} \to [0,1]}

(the intention is that

      P
      
        Y
        
          |
        
        X
      
    
    (
    U
    ,
    x
    )
    =
    P
    (
    Y
    ∈
    U
    
      |
    
    X
    =
    x
    )
  

{\displaystyle P_{Y|X}(U,x)=P(Y\in U|X=x)}

almost surely over the support of X), we can define

    Var
    ⁡
    (
    Y
    
      |
    
    X
    =
    x
    )
    =
    ∫
    
      
        (
        
          y
          −
          ∫
          
            y
            ′
          
          
            P
            
              Y
              
                |
              
              X
            
          
          (
          d
          
            y
            ′
          
          
            |
          
          x
          )
        
        )
      
      
        2
      
    
    
      P
      
        Y
        
          |
        
        X
      
    
    (
    d
    y
    
      |
    
    x
    )
    .
  

{\displaystyle \operatorname {Var} (Y|X=x)=\int \left(y-\int y'P_{Y|X}(dy'|x)\right)^{2}P_{Y|X}(dy|x).}

This can, of course, be specialized to when Y is discrete itself (replacing the integrals with sums), and also when the conditional density of Y given X=x with respect to some underlying distribution exists.

== Components of variance == The law of total variance says

    Var
    ⁡
    (
    Y
    )
    =
    E
    ⁡
    (
    Var
    ⁡
    (
    Y
    ∣
    X
    )
    )
    +
    Var
    ⁡
    (
    E
    ⁡
    (
    Y
    ∣
    X
    )
    )
    .
  

{\displaystyle \operatorname {Var} (Y)=\operatorname {E} (\operatorname {Var} (Y\mid X))+\operatorname {Var} (\operatorname {E} (Y\mid X)).}

In words: the variance of Y is the sum of the expected conditional variance of Y given X and the variance of the conditional expectation of Y given X. The first term captures the variation left after "using X to predict Y", while the second term captures the variation due to the mean of the prediction of Y due to the randomness of X.

== See also == Mixed model Random effects model

== References ==

== Further reading == Casella, George; Berger, Roger L. (2002). Statistical Inference (Second ed.). Wadsworth. pp. 151–52. ISBN 0-534-24312-6.