# Justification of proof of law of total expectation

I am reading about the proof of law of expectation for the discrete type,

\begin{align} E[E[X|Y]] &= E \left[ \sum_{x} x \cdot P(X = x | Y) \right] \\ &= \sum_y \left[ \sum_{x} x \cdot P(X = x | Y = y) \right] P(Y = y) \tag 2) \\ &= \sum_x x \sum_y P(X = x | Y = y) \cdot P(Y = y) \tag3) \\ &= \sum_x x \sum_y P(X = x \, \text{and} \, Y = y) \tag4) \\ &= \sum_x x \cdot P(X = x) \tag5)\\ &= E[X] \end{align}

Question: Why does it assume Y=y? And can someone give me justification of each step?

• This all follows from the definitions of conditional expectation (for discrete random variables), conditional probability, and finding a marginal PMF from a joint PMF. It sounds like you need to review these concepts before trying to understand this proof. Commented Feb 27, 2020 at 18:32
• @Math1000 definitely will do Commented Feb 27, 2020 at 18:46
• At no point is "Y=y" assumed. The only thing involved is the probability of "Y=y". Equalities 1,2,4 and 6 are by definition. Equality 3 is switching the order of summation and equality 5 is additivity of probability for disjoint events. Commented Feb 27, 2020 at 18:53

Question: Why does it assume Y=y? And can someone give me justification of each step?

The second line does not assume $$Y=y$$ any more than the first line assumes $$X=x$$. Both apply the definition of expectation for discrete random variables.

$$\mathsf E(g(Z))~=~\sum_z g(z)\,\mathsf P(Z{=}z)~~\\\mathsf E(h(W)\mid Z{=}z)~=~\sum_w h(w)~\mathsf P(W{=}w\mid Z{=}z)$$

If you prefer, do it from the outside in.

\begin{align}\mathsf E(\mathsf E(X\mid Y))&=\sum_y\mathsf E(X\mid Y{=}y)\,\mathsf P(Y{=}y)\\[1ex]&=\sum_y\left(\sum_x x\,\mathsf P(X{=}x\mid Y{=}y)\right)\mathsf P(Y{=}y)\end{align}

The rest is just distribution, the definition of conditional probability, and the law of total probability.

\begin{align}\phantom{\mathsf E(\mathsf E(X\mid Y))} &=\sum_x\sum_y x\,\mathsf P(X{=}x\mid Y{=}y)\,\mathsf P(Y{=}y)&&{\text{switching order of summation}\\\text{via commutation and association}} \\[2ex] &=\sum_x x\sum_y \mathsf P(X{=}x\mid Y{=}y)\,\mathsf P(Y{=}y)&&\text{distributing out the common factor}\\[1ex]&=\sum_x x\sum_y\mathsf P(X{=}x\cap Y{=}y)&&\text{definition of conditional probability}\\[1ex] &=\sum_x x\,\mathsf P(X{=}x)&&\text{Law of Total Probability}\\[1ex]&=\mathsf E(X)&&\text{definition of expectation}\end{align}

• been looking for a good explanation for hours now and yours is fantastic. Thank you! Commented Feb 6, 2022 at 17:00