Raising the bar Properties of conditional expectation: seven properties in one place

Jul 16

Properties of conditional expectation

category: Agresti & Franklin, Agresti and Franklin, AP Stats and Business Stats, Dougherty Introduction to Econometrics, EC2020, EC2020 Elements of econometrics, Econometrics, FN3142 Quantitative Finance, Quantitative Finance, ST104B, Statistics 2, The College Board 2 Comments

Properties of conditional expectation

Background

A company sells a product and may offer a discount. We denote by $X$ the sales volume and by $Y$ the discount amount (per unit). For simplicity, both variables take only two values. They depend on each other. If the sales are high, the discount may be larger. A higher discount, in its turn, may attract more buyers. At the same level of sales, the discount may vary depending on the vendor's costs. With the same discount, the sales vary with consumer preferences. Along with the sales and discount, we consider a third variable that depends on both of them. It can be the profit $\pi$ .

Formalization

The sales volume $X$ takes values $x_1,x_2$ with probabilities $p_i^X=P(X=x_i)$ , $i=1,2$ . Similarly, the discount $Y$ takes values $y_1,y_2$ with probabilities $p_i^Y=P(Y=y_i)$ , $i=1,2$ . The joint events have joint probabilities denoted $P(X=x_i,Y=y_j)=p_{i,j}$ . The profit in the event $X=x_i,Y=y_j$ is denoted $\pi_{i,j}$ . This information is summarized in Table 1.

Table 1. Values and probabilities of the profit function
	$y_1$	$y_1$
$x_1$	$\pi_{1,1},\ p_{1,1}$	$\pi_{1,2},\ p_{1,2}$	$p_1^X$
$x_2$	$\pi_{2,1},\ p_{2,1}$	$\pi_{2,2},\ p_{2,2}$	$p_2^X$
	$p_1^Y$	$p_2^Y$

Comments. In the left-most column and upper-most row we have values of the sales and discount. In the "margins" (last row and last column) we put probabilities of those values. In the main body of the table we have profit values and their probabilities. It follows that the expected profit is

(1) $E\pi=\pi_{1,1}p_{1,1}+\pi_{1,2}p_{1,2}+\pi_{2,1}p_{2,1}+\pi_{2,2}p_{2,2}.$

Conditioning

Suppose that the vendor fixes the discount at $y_1$ . Then only the column containing this value is relevant. To get numbers that satisfy the completeness axiom, we define conditional probabilities

$P(X=x_1|Y=y_1)=\frac{p_{11}}{p_1^Y},\ P(X=x_2|Y=y_1)=\frac{p_{21}}{p_1^Y}.$

This allows us to define conditional expectation

(2) $E(\pi|Y=y_1)=\pi_{11}\frac{p_{11}}{p_1^Y}+\pi_{21}\frac{p_{21}}{p_1^Y}.$

Similarly, if the discount is fixed at $y_2$ ,

(3) $E(\pi|Y=y_2)=\pi_{12}\frac{p_{12}}{p_2^Y}+\pi_{22}\frac{p_{22}}{p_2^Y}.$

Equations (2) and (3) are joined in the notation $E(\pi|Y)$ .

Property 1. While the usual expectation (1) is a number, the conditional expectation $E(\pi|Y)$ is a function of the value of $Y$ on which the conditioning is being done. Since it is a function of $Y$ , it is natural to consider it a random variable defined by the next table

Table 2. Conditional expectation is a random variable
Values	Probabilities
$E(\pi\|Y=y_1)$	$p_1^Y$
$E(\pi\|Y=y_2)$	$p_2^Y$

Property 2. Law of iterated expectations: the mean of the conditional expectation equals the usual mean. Indeed, using Table 2, we have

$E[E(\pi|Y)]=E(\pi|Y=y_1)p_1^Y+E(\pi|Y=y_2)p_2^Y$ (applying (2) and (3))

$=\left[\pi_{11}\frac{p_{11}}{p_1^Y}+\pi_{21}\frac{p_{21}}{p_1^Y}\right]p_1^Y+\left[\pi_{12}\frac{p_{12}}{p_2^Y}+\pi_{22}\frac{p_{22}}{p_2^Y}\right]p_2^Y$

$=\pi_{1,1}p_{1,1}+\pi_{1,2}p_{1,2}+\pi_{2,1}p_{2,1}+\pi_{2,2}p_{2,2}=E\pi.$

Property 3. Generalized homogeneity. In the usual homogeneity $E(aX)=aEX$ , $a$ is a number. In the generalized homogeneity

(4) $E(a(Y)\pi|Y)=a(Y)E(\pi|Y),$

$a(Y)$ is allowed to be a function of the variable on which we are conditioning. See for yourself: using (2), for instance,

$E(a(y_1)\pi|Y=y_1)=a(y_1)\pi_{11}\frac{p_{11}}{p_1^Y}+a(y_1)\pi_{21}\frac{p_{21}}{p_1^Y}$

$=a(y_1)\left[\pi_{11}\frac{p_{11}}{p_1^Y}+\pi_{21}\frac{p_{21}}{p_1^Y}\right]=a(y_1)E(X|Y=y_1).$

Property 4. Additivity. For any random variables $S,T$ we have

(5) $E(S+T|Y)=E(S|Y)+E(T|Y).$

The proof is left as an exercise.

Property 5. Generalized linearity. For any random variables $S,T$ and functions $a(Y),b(Y)$ equations (4) and (5) imply

$E(a(Y)S+b(Y)T|Y)=a(Y)E(S|Y)+b(Y)E(T|Y).$

Property 6. Conditioning in case of independence. This property has to do with the informational aspect of conditioning. The usual expectation (1) takes into account all contingencies. (2) and (3) are based on the assumption that one contingency for $Y$ has been realized, so that the other one becomes irrelevant. Therefore $E(\pi|Y)$ is considered an updated version of (1) that takes into account the arrival of new information that the value of $Y$ has been fixed. Now we can state the property itself: if $X,Y$ are independent, then $E(X|Y)=EX$ , that is, conditioning on $Y$ does not improve our knowledge of $EX$ .

Proof. In case of independence we have $p_{i,j}=p_i^Xp_j^Y$ for all $i,j$ , so that

$E(X|Y=y_j)=x_1\frac{p_{1j}}{p_j^Y}+x_2\frac{p_{2j}}{p_j^Y}=x_1p_1^X+x_2p_2^X=EX.$

Property 7. Conditioning in case of complete dependence. Conditioning of $Y$ on $Y$ gives the most precise information: $E(Y|Y)=Y$ (if we condition $Y$ on $Y$ , we know about it everything and there is no averaging). More generally, $E(f(Y)|Y)=f(Y)$ for any deterministic function $f$ .

Proof. If we condition $Y$ on $Y$ , the conditional probabilities become

$p_{11}=P(Y=y_1|Y=y_1)=1,\ p_{21}=P(Y=y_2|Y=y_1)=0.$

Hence, (2) gives

$E(f(Y)|Y=y_1)=f(y_1)\times 1+f(y_2)\times 0=f(y_1).$

Conditioning on $Y=y_2$ is treated similarly.

Summary

Not many people know that using the notation $E_Y\pi$ for conditional expectation instead of $E(\pi|Y)$ makes everything much clearer. I rewrite the above properties using this notation:

Law of iterated expectations: $E(E_Y\pi)=E\pi$
Generalized homogeneity: $E_Y(a(Y)\pi)=a(Y)E_Y\pi$
Additivity: For any random variables $S,T$ we have $E_Y(S+T)=E_YS+E_YT$
Generalized linearity: For any random variables $S,T$ and functions $a(Y),b(Y)$ one has $E_Y(a(Y)S+b(Y)T)=a(Y)E_YS+b(Y)E_YT$
Conditioning in case of independence: if $X,Y$ are independent, then $E_YX=EX$
Conditioning in case of complete dependence: $E_Yf(Y)=f(Y)$ for any deterministic function $f$ .