Question

Conditional Expectation If X and Y are both discrete random variables, we have $E[X | Y] = sum_{x_j} x_j p_x(x_j | y)$ We now show that $E[X] = E[E[X | Y]]$ (Law of total expectation) $E_Y(E_{X|Y}(X | Y)) = E_Y left[ sum_x xP(X = x | Y) ight] = sum_y left[ sum_x xP(X = x | Y = y) ight] P(Y = y)$ $= sum_y sum_x xP(X = x | Y = y)P(Y = y) = sum_x x sum_y P(X = x | Y = y)P(Y = y)$ $= sum_x x sum_y P(X = x, Y = y) = sum_x xP(X = x) = E(X)$

          Conditional Expectation
If X and Y are both discrete random variables, we have
$E[X | Y] = sum_{x_j} x_j p_x(x_j | y)$
We now show that
$E[X] = E[E[X | Y]]$ (Law of total expectation)
$E_Y(E_{X|Y}(X | Y)) = E_Y left[ sum_x xP(X = x | Y) 
ight] = sum_y left[ sum_x xP(X = x | Y = y) 
ight] P(Y = y)$
$= sum_y sum_x xP(X = x | Y = y)P(Y = y) = sum_x x sum_y P(X = x | Y = y)P(Y = y)$
$= sum_x x sum_y P(X = x, Y = y) = sum_x xP(X = x) = E(X)$

Conditional Expectation
If X and Y are both discrete random variables, we have
E[X | Y] = sumxj xj px(xj | y)
We now show that
E[X] = E[E[X | Y]] (Law of total expectation)
EY(EX|Y(X | Y)) = EY left[ sumx xP(X = x | Y)
ight] = sumy left[ sumx xP(X = x | Y = y)
ight] P(Y = y)
= sumy sumx xP(X = x | Y = y)P(Y = y) = sumx x sumy P(X = x | Y = y)P(Y = y)
= sumx x sumy P(X = x, Y = y) = sumx xP(X = x) = E(X)

Added by Gabriel Q.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Shaiju T

04/06/2022

Step 1

The conditional expectation of X given Y is defined as: $E[X|Y] = \sum_{x_j} x_j p_x(x_j|y)$ where $x_j$ are the possible values of X and $p_x(x_j|y)$ is the conditional probability mass function of X given Y. Show more…

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Conditional Expectation If X and Y are both discrete random variables, we have $E[X|Y] = sum_{x_j} x_j p_x(x_j|y)$ We now show that $E[X] = E[E[X|Y]]$ (Law of total expectation) $E_Y(E_{X|Y}(X|Y)) = E_Y[sum_x xP(X=x|Y)] = sum_y [sum_x xP(X=x|Y=y)]P(Y=y)$ $= sum_y sum_x xP(X=x|Y=y)P(Y=y) = sum_x sum_y xP(X=x|Y=y)P(Y=y)$ $= sum_x x sum_y P(X=x, Y=y) = sum_x xP(X=x) = E(X)$