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