Conditional expectation

Consider the roll of a fair die and let A = 1 if the number is even (i.e., 2, 4, or 6) and A = 0 otherwise. Furthermore, let B = 1 if the number is prime (i.e., 2, 3, or 5) and B = 0 otherwise.

The unconditional expectation of A is ${\displaystyle E[$A]=(0+1+0+1+0+1)/6=1/2}  , but the expectation of A conditional on B = 1 (i.e., conditional on the die roll being 2, 3, or 5) is ${\displaystyle E[A\mid B=1]=(1+0+0)/3=1/3}$ , and the expectation of A conditional on B = 0 (i.e., conditional on the die roll being 1, 4, or 6) is ${\displaystyle E[A\mid B=0]=(0+1+1)/3=2/3}$ . Likewise, the expectation of B conditional on A = 1 is ${\displaystyle E[B\mid A=1]=(1+0+0)/3=1/3}$ , and the expectation of B conditional on A = 0 is ${\displaystyle E[B\mid A=0]=(0+1+1)/3=2/3}$ .

Suppose we have daily rainfall data (mm of rain each day) collected by a weather station on every day of the ten–year (3652-day) period from January 1, 1990, to December 31, 1999. The unconditional expectation of rainfall for an unspecified day is the average of the rainfall amounts for those 3652 days. The conditional expectation of rainfall for an otherwise unspecified day known to be (conditional on being) in the month of March, is the average of daily rainfall over all 310 days of the ten–year period that falls in March. And the conditional expectation of rainfall conditional on days dated March 2 is the average of the rainfall amounts that occurred on the ten days with that specific date.

The related concept of conditional probability dates back at least to Laplace, who calculated conditional distributions. It was Andrey Kolmogorov who, in 1933, formalized it using the Radon–Nikodym theorem.^[1] In works of Paul Halmos^[2] and Joseph L. Doob^[3] from 1953, conditional expectation was generalized to its modern definition using sub-σ-algebras.^[4]

IfA is an event in ${\displaystyle {\mathcal {F}}}$  with nonzero probability, and X is a discrete random variable, the conditional expectation of X given Ais

${\displaystyle {\begin{aligned}\operatorname {E} (X\mid A)&=\sum _{x}xP(X=x\mid A)\\&=\sum _{x}x{\frac {P(\{X=x\}\cap A)}{P($A)}}\end{aligned}}}  

where the sum is taken over all possible outcomes of X.

If${\displaystyle P($A)=0}  , the conditional expectation is undefined due to the division by zero.

IfX and Y are discrete random variables, the conditional expectation of X given Yis

${\displaystyle {\begin{aligned}\operatorname {E} (X\mid Y=y)&=\sum _{x}xP(X=x\mid Y=y)\\&=\sum _{x}x{\frac {P(X=x,Y=y)}{P(Y=y)}}\end{aligned}}}$ 

where ${\displaystyle P(X=x,Y=y)}$  is the joint probability mass functionofX and Y. The sum is taken over all possible outcomes of X.

Remark that as above the expression is undefined if ${\displaystyle P(Y=y)=0}$ .

Conditioning on a discrete random variable is the same as conditioning on the corresponding event:

${\displaystyle \operatorname {E} (X\mid Y=y)=\operatorname {E} (X\mid A)}$ 

where A is the set ${\displaystyle \{Y=y\}}$ .

Let ${\displaystyle X}$  and ${\displaystyle Y}$ becontinuous random variables with joint density ${\displaystyle f_{X,Y}(x,y),}$  ${\displaystyle Y}$ 's density ${\displaystyle f_{Y}($y),}   and conditional density ${\displaystyle \textstyle f_{X|Y}(x|y)={\frac {f_{X,Y}(x,y)}{f_{Y}($y)}}}  of${\displaystyle X}$  given the event ${\displaystyle Y=y.}$  The conditional expectation of ${\displaystyle X}$  given ${\displaystyle Y=y}$ is

${\displaystyle {\begin{aligned}\operatorname {E} (X\mid Y=y)&=\int _{-\infty }^{\infty }xf_{X|Y}(x\mid y)\,\mathrm {d} x\\&={\frac {1}{f_{Y}($y)}}\int _{-\infty }^{\infty }xf_{X,Y}(x,y)\,\mathrm {d} x.\end{aligned}}}  

When the denominator is zero, the expression is undefined.

Conditioning on a continuous random variable is not the same as conditioning on the event ${\displaystyle \{Y=y\}}$  as it was in the discrete case. For a discussion, see Conditioning on an event of probability zero. Not respecting this distinction can lead to contradictory conclusions as illustrated by the Borel-Kolmogorov paradox.

All random variables in this section are assumed to be in ${\displaystyle L^{2}}$ , that is square integrable. In its full generality, conditional expectation is developed without this assumption, see below under Conditional expectation with respect to a sub-σ-algebra. The ${\displaystyle L^{2}}$  theory is, however, considered more intuitive^[5] and admits important generalizations. In the context of ${\displaystyle L^{2}}$  random variables, conditional expectation is also called regression.

In what follows let ${\displaystyle (\Omega ,{\mathcal {F}},P)}$  be a probability space, and ${\displaystyle X:\Omega \to \mathbb {R} }$ in${\displaystyle L^{2}}$  with mean ${\displaystyle \mu _{X}}$  and variance ${\displaystyle \sigma _{X}^{2}}$ . The expectation ${\displaystyle \mu _{X}}$  minimizes the mean squared error:

${\displaystyle \min _{x\in \mathbb {R} }\operatorname {E} \left((X-x)^{2}\right)=\operatorname {E} \left((X-\mu _{X})^{2}\right)=\sigma _{X}^{2}}$ .

The conditional expectation of X is defined analogously, except instead of a single number ${\displaystyle \mu _{X}}$ , the result will be a function ${\displaystyle e_{X}($y)}  . Let ${\displaystyle Y:\Omega \to \mathbb {R} ^{n}}$  be a random vector. The conditional expectation ${\displaystyle e_{X}:\mathbb {R} ^{n}\to \mathbb {R} }$  is a measurable function such that

${\displaystyle \min _{g{\text{ measurable }}}\operatorname {E} \left((X-g($Y))^{2}\right)=\operatorname {E} \left((X-e_{X}(Y))^{2}\right)}  .

Note that unlike ${\displaystyle \mu _{X}}$ , the conditional expectation ${\displaystyle e_{X}}$  is not generally unique: there may be multiple minimizers of the mean squared error.

Example 1: Consider the case where Y is the constant random variable that's always 1. Then the mean squared error is minimized by any function of the form

${\displaystyle e_{X}($y)={\begin{cases}\mu _{X}&{\text{ if }}y=1\\{\text{any number}}&{\text{ otherwise}}\end{cases}}}  

Example 2: Consider the case where Y is the 2-dimensional random vector ${\displaystyle (X,2X)}$ . Then clearly

${\displaystyle \operatorname {E} (X\mid Y)=X}$ 

but in terms of functions it can be expressed as ${\displaystyle e_{X}(y_{1},y_{2})=3y_{1}-y_{2}}$ or${\displaystyle e'_{X}(y_{1},y_{2})=y_{2}-y_{1}}$  or infinitely many other ways. In the context of linear regression, this lack of uniqueness is called multicollinearity.

Conditional expectation is unique up to a set of measure zero in ${\displaystyle \mathbb {R} ^{n}}$ . The measure used is the pushforward measure induced by Y.

In the first example, the pushforward measure is a Dirac distribution at 1. In the second it is concentrated on the "diagonal" ${\displaystyle \{y:y_{2}=2y_{1}\}}$ , so that any set not intersecting it has measure 0.

The existence of a minimizer for ${\displaystyle \min _{g}\operatorname {E} \left((X-g($Y))^{2}\right)}   is non-trivial. It can be shown that

${\displaystyle M:=\{g($Y):g{\text{ is measurable and }}\operatorname {E} (g(Y)^{2})<\infty \}=L^{2}(\Omega ,\sigma (Y))}  

is a closed subspace of the Hilbert space ${\displaystyle L^{2}(\Omega )}$ .^[6] By the Hilbert projection theorem, the necessary and sufficient condition for ${\displaystyle e_{X}}$  to be a minimizer is that for all ${\displaystyle f($Y)}  inM we have

${\displaystyle \langle X-e_{X}($Y),f(Y)\rangle =0}  .

In words, this equation says that the residual ${\displaystyle X-e_{X}($Y)}   is orthogonal to the space M of all functions of Y. This orthogonality condition, applied to the indicator functions ${\displaystyle f($Y)=1_{Y\in H}}  , is used below to extend conditional expectation to the case that X and Y are not necessarily in ${\displaystyle L^{2}}$ .

The conditional expectation is often approximated in applied mathematics and statistics due to the difficulties in analytically calculating it, and for interpolation.^[7]

The Hilbert subspace

${\displaystyle M=\{g($Y):\operatorname {E} (g(Y)^{2})<\infty \}}  

defined above is replaced with subsets thereof by restricting the functional form of g, rather than allowing any measurable function. Examples of this are decision tree regression when g is required to be a simple function, linear regression when g is required to be affine, etc.

These generalizations of conditional expectation come at the cost of many of its properties no longer holding. For example, let M be the space of all linear functions of Y and let ${\displaystyle {\mathcal {E}}_{M}}$  denote this generalized conditional expectation/${\displaystyle L^{2}}$  projection. If ${\displaystyle M}$  does not contain the constant functions, the tower property ${\displaystyle \operatorname {E} ({\mathcal {E}}_{M}($X))=\operatorname {E} (X)}   will not hold.

An important special case is when X and Y are jointly normally distributed. In this case it can be shown that the conditional expectation is equivalent to linear regression:

${\displaystyle e_{X}($Y)=\alpha _{0}+\sum _{i}\alpha _{i}Y_{i}}  

for coefficients ${\displaystyle \{\alpha _{i}\}_{i=0..n}}$  described in Multivariate normal distribution#Conditional distributions.

Consider the following:

• ${\displaystyle (\Omega ,{\mathcal {F}},P)}$  is a probability space.
• ${\displaystyle X\colon \Omega \to \mathbb {R} ^{n}}$  is a random variable on that probability space with finite expectation.
• ${\displaystyle {\mathcal {H}}\subseteq {\mathcal {F}}}$  is a sub-σ-algebraof${\displaystyle {\mathcal {F}}}$ .

Since ${\displaystyle {\mathcal {H}}}$  is a sub ${\displaystyle \sigma }$ -algebra of ${\displaystyle {\mathcal {F}}}$ , the function ${\displaystyle X\colon \Omega \to \mathbb {R} ^{n}}$  is usually not ${\displaystyle {\mathcal {H}}}$ -measurable, thus the existence of the integrals of the form ${\textstyle \int _{H}X\,dP|_{\mathcal {H}}}$ , where ${\displaystyle H\in {\mathcal {H}}}$  and ${\displaystyle P|_{\mathcal {H}}}$  is the restriction of ${\displaystyle P}$ to${\displaystyle {\mathcal {H}}}$ , cannot be stated in general. However, the local averages ${\textstyle \int _{H}X\,dP}$  can be recovered in ${\displaystyle (\Omega ,{\mathcal {H}},P|_{\mathcal {H}})}$  with the help of the conditional expectation.

Aconditional expectationofX given ${\displaystyle {\mathcal {H}}}$ , denoted as ${\displaystyle \operatorname {E} (X\mid {\mathcal {H}})}$ , is any ${\displaystyle {\mathcal {H}}}$ -measurable function ${\displaystyle \Omega \to \mathbb {R} ^{n}}$  which satisfies:

${\displaystyle \int _{H}\operatorname {E} (X\mid {\mathcal {H}})\,\mathrm {d} P=\int _{H}X\,\mathrm {d} P}$ 

for each ${\displaystyle H\in {\mathcal {H}}}$ .^[8]

As noted in the ${\displaystyle L^{2}}$  discussion, this condition is equivalent to saying that the residual ${\displaystyle X-\operatorname {E} (X\mid {\mathcal {H}})}$  is orthogonal to the indicator functions ${\displaystyle 1_{H}}$ :

${\displaystyle \langle X-\operatorname {E} (X\mid {\mathcal {H}}),1_{H}\rangle =0}$ 

The existence of ${\displaystyle \operatorname {E} (X\mid {\mathcal {H}})}$  can be established by noting that ${\textstyle \mu ^{X}\colon F\mapsto \int _{F}X\,\mathrm {d} P}$  for ${\displaystyle F\in {\mathcal {F}}}$  is a finite measure on ${\displaystyle (\Omega ,{\mathcal {F}})}$  that is absolutely continuous with respect to ${\displaystyle P}$ . If ${\displaystyle h}$  is the natural injection from ${\displaystyle {\mathcal {H}}}$ to${\displaystyle {\mathcal {F}}}$ , then ${\displaystyle \mu ^{X}\circ h=\mu ^{X}|_{\mathcal {H}}}$  is the restriction of ${\displaystyle \mu ^{X}}$ to${\displaystyle {\mathcal {H}}}$  and ${\displaystyle P\circ h=P|_{\mathcal {H}}}$  is the restriction of ${\displaystyle P}$ to${\displaystyle {\mathcal {H}}}$ . Furthermore, ${\displaystyle \mu ^{X}\circ h}$  is absolutely continuous with respect to ${\displaystyle P\circ h}$ , because the condition

${\displaystyle P\circ h($H)=0\iff P(h(H))=0}  

implies

${\displaystyle \mu ^{X}(h($H))=0\iff \mu ^{X}\circ h(H)=0.}  

Thus, we have

${\displaystyle \operatorname {E} (X\mid {\mathcal {H}})={\frac {\mathrm {d} \mu ^{X}|_{\mathcal {H}}}{\mathrm {d} P|_{\mathcal {H}}}}={\frac {\mathrm {d} (\mu ^{X}\circ h)}{\mathrm {d} (P\circ h)}},}$ 

where the derivatives are Radon–Nikodym derivatives of measures.

Consider, in addition to the above,

• Ameasurable space ${\displaystyle (U,\Sigma )}$ , and
• A random variable ${\displaystyle Y\colon \Omega \to U}$ .

The conditional expectation of X given Y is defined by applying the above construction on the σ-algebra generated by Y:

${\displaystyle \operatorname {E} [X\mid Y]:=\operatorname {E} [X\mid \sigma ($Y)]}  .

By the Doob-Dynkin lemma, there exists a function ${\displaystyle e_{X}\colon U\to \mathbb {R} ^{n}}$  such that

${\displaystyle \operatorname {E} [X\mid Y]=e_{X}($Y)}  .

• This is not a constructive definition; we are merely given the required property that a conditional expectation must satisfy.
  • The definition of ${\displaystyle \operatorname {E} (X\mid {\mathcal {H}})}$  may resemble that of ${\displaystyle \operatorname {E} (X\mid H)}$  for an event ${\displaystyle H}$  but these are very different objects. The former is a ${\displaystyle {\mathcal {H}}}$ -measurable function ${\displaystyle \Omega \to \mathbb {R} ^{n}}$ , while the latter is an element of ${\displaystyle \mathbb {R} ^{n}}$  and ${\displaystyle \operatorname {E} (X\mid H)\ P($H)=\int _{H}X\,\mathrm {d} P=\int _{H}\operatorname {E} (X\mid {\mathcal {H}})\,\mathrm {d} P}   for ${\displaystyle H\in {\mathcal {H}}}$ .
  • Uniqueness can be shown to be almost sure: that is, versions of the same conditional expectation will only differ on a set of probability zero.
• The σ-algebra ${\displaystyle {\mathcal {H}}}$  controls the "granularity" of the conditioning. A conditional expectation ${\displaystyle E(X\mid {\mathcal {H}})}$  over a finer (larger) σ-algebra ${\displaystyle {\mathcal {H}}}$  retains information about the probabilities of a larger class of events. A conditional expectation over a coarser (smaller) σ-algebra averages over more events.

For a Borel subset Bin${\displaystyle {\mathcal {B}}(\mathbb {R} ^{n})}$ , one can consider the collection of random variables

${\displaystyle \kappa _{\mathcal {H}}(\omega ,B):=\operatorname {E} (1_{X\in B}|{\mathcal {H}})(\omega )}$ .

It can be shown that they form a Markov kernel, that is, for almost all ${\displaystyle \omega }$ , ${\displaystyle \kappa _{\mathcal {H}}(\omega ,-)}$  is a probability measure.^[9]

The Law of the unconscious statistician is then

${\displaystyle \operatorname {E} [f($X)|{\mathcal {H}}]=\int f(x)\kappa _{\mathcal {H}}(-,\mathrm {d} x)}  .

This shows that conditional expectations are, like their unconditional counterparts, integrations, against a conditional measure.

In full generality, consider:

• A probability space ${\displaystyle (\Omega ,{\mathcal {A}},P)}$ .
• ABanach space ${\displaystyle (E,\|\cdot \|_{E})}$ .
• ABochner integrable random variable ${\displaystyle X:\Omega \to E}$ .
• A sub-σ-algebra ${\displaystyle {\mathcal {H}}\subseteq {\mathcal {A}}}$ .

The conditional expectationof${\displaystyle X}$  given ${\displaystyle {\mathcal {H}}}$  is the up to a ${\displaystyle P}$ -nullset unique and integrable ${\displaystyle E}$ -valued ${\displaystyle {\mathcal {H}}}$ -measurable random variable ${\displaystyle \operatorname {E} (X\mid {\mathcal {H}})}$  satisfying

${\displaystyle \int _{H}\operatorname {E} (X\mid {\mathcal {H}})\,\mathrm {d} P=\int _{H}X\,\mathrm {d} P}$ 

for all ${\displaystyle H\in {\mathcal {H}}}$ .^[10][11]

In this setting the conditional expectation is sometimes also denoted in operator notation as ${\displaystyle \operatorname {E} ^{\mathcal {H}}X}$ .

All the following formulas are to be understood in an almost sure sense. The σ-algebra ${\displaystyle {\mathcal {H}}}$  could be replaced by a random variable ${\displaystyle Z}$ , i.e. ${\displaystyle {\mathcal {H}}=\sigma ($Z)}  .

• Pulling out independent factors:
  • If${\displaystyle X}$ isindependentof${\displaystyle {\mathcal {H}}}$ , then ${\displaystyle E(X\mid {\mathcal {H}})=E($X)}  .

${\displaystyle \int _{B}X\,dP=E(X1_{B})=E($X)E(1_{B})=E(X)P(B)=\int _{B}E(X)\,dP.}  

Thus the definition of conditional expectation is satisfied by the constant random variable ${\displaystyle E($X)}  , as desired. ${\displaystyle \square }$ 

• • If${\displaystyle X}$  is independent of ${\displaystyle \sigma (Y,{\mathcal {H}})}$ , then ${\displaystyle E(XY\mid {\mathcal {H}})=E($X)\,E(Y\mid {\mathcal {H}})}  . Note that this is not necessarily the case if ${\displaystyle X}$  is only independent of ${\displaystyle {\mathcal {H}}}$  and of ${\displaystyle Y}$ .
  • If${\displaystyle X,Y}$  are independent, ${\displaystyle {\mathcal {G}},{\mathcal {H}}}$  are independent, ${\displaystyle X}$  is independent of ${\displaystyle {\mathcal {H}}}$  and ${\displaystyle Y}$  is independent of ${\displaystyle {\mathcal {G}}}$ , then ${\displaystyle E(E(XY\mid {\mathcal {G}})\mid {\mathcal {H}})=E($X)E(Y)=E(E(XY\mid {\mathcal {H}})\mid {\mathcal {G}})}  .
• Stability:
  • If${\displaystyle X}$ is${\displaystyle {\mathcal {H}}}$ -measurable, then ${\displaystyle E(X\mid {\mathcal {H}})=X}$ .

${\displaystyle \int _{H}{\big (}E(X|{\mathcal {H}})-X{\big )}dP=0}$ 

Since this is true for each ${\displaystyle H\in {\mathcal {H}}}$ , and both ${\displaystyle E(X|{\mathcal {H}})}$  and ${\displaystyle X}$  are ${\displaystyle {\mathcal {H}}}$ -measurable (the former property holds by definition; the latter property is key here), from this one can show

${\displaystyle \int _{H}{\big |}E(X|{\mathcal {H}})-X{\big |}dP=0}$ 

And this implies ${\displaystyle E(X|{\mathcal {H}})=X}$  almost everywhere. ${\displaystyle \square }$ 

• • In particular, for sub-σ-algebras ${\displaystyle {\mathcal {H}}_{1}\subset {\mathcal {H}}_{2}\subset {\mathcal {F}}}$  we have ${\displaystyle E(E(X\mid {\mathcal {H}}_{1})\mid {\mathcal {H}}_{2})=E(X\mid {\mathcal {H}}_{1})}$ .
  • IfZ is a random variable, then ${\displaystyle \operatorname {E} (f($Z)\mid Z)=f(Z)}  . In its simplest form, this says ${\displaystyle \operatorname {E} (Z\mid Z)=Z}$ .
• Pulling out known factors:
  • If${\displaystyle X}$ is${\displaystyle {\mathcal {H}}}$ -measurable, then ${\displaystyle E(XY\mid {\mathcal {H}})=X\,E(Y\mid {\mathcal {H}})}$ .

Fix ${\displaystyle A\in {\mathcal {H}}}$  and let ${\displaystyle X=1_{A}}$ . Then for any ${\displaystyle H\in {\mathcal {H}}}$ 

${\displaystyle \int _{H}E(1_{A}Y|{\mathcal {H}})dP=\int _{H}1_{A}YdP=\int _{A\cap H}YdP=\int _{A\cap H}E(Y|{\mathcal {H}})dP=\int _{H}1_{A}E(Y|{\mathcal {H}})dP}$ 

Hence ${\displaystyle E(1_{A}Y|{\mathcal {H}})=1_{A}E(Y|{\mathcal {H}})}$  almost everywhere.

Any simple function is a finite linear combination of indicator functions. By linearity the above property holds for simple functions: if ${\displaystyle X_{n}}$  is a simple function then ${\displaystyle E(X_{n}Y|{\mathcal {H}})=X_{n}\,E(Y|{\mathcal {H}})}$ .

Now let ${\displaystyle X}$ be${\displaystyle {\mathcal {H}}}$ -measurable. Then there exists a sequence of simple functions ${\displaystyle \{X_{n}\}_{n\geq 1}}$  converging monotonically (here meaning ${\displaystyle X_{n}\leq X_{n+1}}$ ) and pointwise to ${\displaystyle X}$ . Consequently, for ${\displaystyle Y\geq 0}$ , the sequence ${\displaystyle \{X_{n}Y\}_{n\geq 1}}$  converges monotonically and pointwise to ${\displaystyle XY}$ .

Also, since ${\displaystyle E(Y|{\mathcal {H}})\geq 0}$ , the sequence ${\displaystyle \{X_{n}E(Y|{\mathcal {H}})\}_{n\geq 1}}$  converges monotonically and pointwise to ${\displaystyle X\,E(Y|{\mathcal {H}})}$ 

Combining the special case proved for simple functions, the definition of conditional expectation, and deploying the monotone convergence theorem:

${\displaystyle \int _{H}X\,E(Y|{\mathcal {H}})dP=\int _{H}\lim _{n\to \infty }X_{n}\,E(Y|{\mathcal {H}})dP=\lim _{n\to \infty }\int _{H}X_{n}E(Y|{\mathcal {H}})dP=\lim _{n\to \infty }\int _{H}E(X_{n}Y|{\mathcal {H}})dP=\lim _{n\to \infty }\int _{H}X_{n}YdP=\int _{H}\lim _{n\to \infty }X_{n}YdP=\int _{H}XYdP=\int _{H}E(XY|{\mathcal {H}})dP}$ 

This holds for all ${\displaystyle H\in {\mathcal {H}}}$ , whence ${\displaystyle X\,E(Y|{\mathcal {H}})=E(XY|{\mathcal {H}})}$  almost everywhere. ${\displaystyle \square }$ 

• • IfZ is a random variable, then ${\displaystyle \operatorname {E} (f($Z)Y\mid Z)=f(Z)\operatorname {E} (Y\mid Z)}  .
• Law of total expectation: ${\displaystyle E(E(X\mid {\mathcal {H}}))=E($X)}  .^[12]
• Tower property:
  • For sub-σ-algebras ${\displaystyle {\mathcal {H}}_{1}\subset {\mathcal {H}}_{2}\subset {\mathcal {F}}}$  we have ${\displaystyle E(E(X\mid {\mathcal {H}}_{2})\mid {\mathcal {H}}_{1})=E(X\mid {\mathcal {H}}_{1})}$ .
    • A special case ${\displaystyle {\mathcal {H}}_{1}=\{\emptyset ,\Omega \}}$  recovers the Law of total expectation: ${\displaystyle E(E(X\mid {\mathcal {H}}_{2}))=E($X)}  .
    • A special case is when Z is a ${\displaystyle {\mathcal {H}}}$ -measurable random variable. Then ${\displaystyle \sigma ($Z)\subset {\mathcal {H}}}   and thus ${\displaystyle E(E(X\mid {\mathcal {H}})\mid Z)=E(X\mid Z)}$ .
    • Doob martingale property: the above with ${\displaystyle Z=E(X\mid {\mathcal {H}})}$  (which is ${\displaystyle {\mathcal {H}}}$ -measurable), and using also ${\displaystyle \operatorname {E} (Z\mid Z)=Z}$ , gives ${\displaystyle E(X\mid E(X\mid {\mathcal {H}}))=E(X\mid {\mathcal {H}})}$ .
  • For random variables ${\displaystyle X,Y}$  we have ${\displaystyle E(E(X\mid Y)\mid f($Y))=E(X\mid f(Y))}  .
  • For random variables ${\displaystyle X,Y,Z}$  we have ${\displaystyle E(E(X\mid Y,Z)\mid Y)=E(X\mid Y)}$ .
• Linearity: we have ${\displaystyle E(X_{1}+X_{2}\mid {\mathcal {H}})=E(X_{1}\mid {\mathcal {H}})+E(X_{2}\mid {\mathcal {H}})}$  and ${\displaystyle E(aX\mid {\mathcal {H}})=a\,E(X\mid {\mathcal {H}})}$  for ${\displaystyle a\in \mathbb {R} }$ .
• Positivity: If ${\displaystyle X\geq 0}$  then ${\displaystyle E(X\mid {\mathcal {H}})\geq 0}$ .
• Monotonicity: If ${\displaystyle X_{1}\leq X_{2}}$  then ${\displaystyle E(X_{1}\mid {\mathcal {H}})\leq E(X_{2}\mid {\mathcal {H}})}$ .
• Monotone convergence: If ${\displaystyle 0\leq X_{n}\uparrow X}$  then ${\displaystyle E(X_{n}\mid {\mathcal {H}})\uparrow E(X\mid {\mathcal {H}})}$ .
• Dominated convergence: If ${\displaystyle X_{n}\to X}$  and ${\displaystyle |X_{n}|\leq Y}$  with ${\displaystyle Y\in L^{1}}$ , then ${\displaystyle E(X_{n}\mid {\mathcal {H}})\to E(X\mid {\mathcal {H}})}$ .
• Fatou's lemma: If ${\displaystyle \textstyle E(\inf _{n}X_{n}\mid {\mathcal {H}})>-\infty }$  then ${\displaystyle \textstyle E(\liminf _{n\to \infty }X_{n}\mid {\mathcal {H}})\leq \liminf _{n\to \infty }E(X_{n}\mid {\mathcal {H}})}$ .
• Jensen's inequality: If ${\displaystyle f\colon \mathbb {R} \rightarrow \mathbb {R} }$  is a convex function, then ${\displaystyle f(E(X\mid {\mathcal {H}}))\leq E(f($X)\mid {\mathcal {H}})}  .
• Conditional variance: Using the conditional expectation we can define, by analogy with the definition of the variance as the mean square deviation from the average, the conditional variance
  • Definition: ${\displaystyle \operatorname {Var} (X\mid {\mathcal {H}})=\operatorname {E} {\bigl (}(X-\operatorname {E} (X\mid {\mathcal {H}}))^{2}\mid {\mathcal {H}}{\bigr )}}$ 
  • Algebraic formula for the variance: ${\displaystyle \operatorname {Var} (X\mid {\mathcal {H}})=\operatorname {E} (X^{2}\mid {\mathcal {H}})-{\bigl (}\operatorname {E} (X\mid {\mathcal {H}}){\bigr )}^{2}}$ 
  • Law of total variance: ${\displaystyle \operatorname {Var} ($X)=\operatorname {E} (\operatorname {Var} (X\mid {\mathcal {H}}))+\operatorname {Var} (\operatorname {E} (X\mid {\mathcal {H}}))}  .
• Martingale convergence: For a random variable ${\displaystyle X}$ , that has finite expectation, we have ${\displaystyle E(X\mid {\mathcal {H}}_{n})\to E(X\mid {\mathcal {H}})}$ , if either ${\displaystyle {\mathcal {H}}_{1}\subset {\mathcal {H}}_{2}\subset \dotsb }$  is an increasing series of sub-σ-algebras and ${\displaystyle \textstyle {\mathcal {H}}=\sigma (\bigcup _{n=1}^{\infty }{\mathcal {H}}_{n})}$  or if ${\displaystyle {\mathcal {H}}_{1}\supset {\mathcal {H}}_{2}\supset \dotsb }$  is a decreasing series of sub-σ-algebras and ${\displaystyle \textstyle {\mathcal {H}}=\bigcap _{n=1}^{\infty }{\mathcal {H}}_{n}}$ .
• Conditional expectation as ${\displaystyle L^{2}}$ -projection: If ${\displaystyle X,Y}$  are in the Hilbert spaceofsquare-integrable real random variables (real random variables with finite second moment) then
  • for ${\displaystyle {\mathcal {H}}}$ -measurable ${\displaystyle Y}$ , we have ${\displaystyle E(Y(X-E(X\mid {\mathcal {H}})))=0}$ , i.e. the conditional expectation ${\displaystyle E(X\mid {\mathcal {H}})}$  is in the sense of the L²(P) scalar product the orthogonal projection from ${\displaystyle X}$  to the linear subspaceof${\displaystyle {\mathcal {H}}}$ -measurable functions. (This allows to define and prove the existence of the conditional expectation based on the Hilbert projection theorem.)
  • the mapping ${\displaystyle X\mapsto \operatorname {E} (X\mid {\mathcal {H}})}$ isself-adjoint: ${\displaystyle \operatorname {E} (X\operatorname {E} (Y\mid {\mathcal {H}}))=\operatorname {E} \left(\operatorname {E} (X\mid {\mathcal {H}})\operatorname {E} (Y\mid {\mathcal {H}})\right)=\operatorname {E} (\operatorname {E} (X\mid {\mathcal {H}})Y)}$ 
• Conditioning is a contractive projection of L^p spaces ${\displaystyle L^{p}(\Omega ,{\mathcal {F}},P)\rightarrow L^{p}(\Omega ,{\mathcal {H}},P)}$ . I.e., ${\displaystyle \operatorname {E} {\big (}|\operatorname {E} (X\mid {\mathcal {H}})|^{p}{\big )}\leq \operatorname {E} {\big (}|X|^{p}{\big )}}$  for any p ≥ 1.
• Doob's conditional independence property:^[13]If${\displaystyle X,Y}$  are conditionally independent given ${\displaystyle Z}$ , then ${\displaystyle P(X\in B\mid Y,Z)=P(X\in B\mid Z)}$  (equivalently, ${\displaystyle E(1_{\{X\in B\}}\mid Y,Z)=E(1_{\{X\in B\}}\mid Z)}$ ).