0

Picture of the Task here

I got this task to solve and i am very dissapointet of myself that i can't solve this. I will write only m instead of Municipality and F instead of FederlState.

For a) i got $$\mathbb{E}[wage\vert m]=2 \mathbb{1}_{m=1}+\mathbb{1}_{m=2}+2.5\mathbb{1}_{m=3}+2.5\mathbb{1}_{m=4}. $$ For b) i have $$ \mathbb{E}[\vert F]=\frac{9}{4} \mathbb{1}_{F=1}+2 \mathbb{1}_{F=2}. $$ So nowc) For the first one i have $$ \mathbb{E}[\mathbb{E} [wage \vert m]\vert F] \overset{tower\; property}{=} \mathbb{E}[wage\vert F]=\frac{9}{4} \mathbb{1}_{F=1}+2 \mathbb{1}_{F=2},$$ since $\sigma(F)\subset \sigma(m)$. For the second one i am not quite sure what to do. My idea was $$ \mathbb{E} [\mathbb{E} [wage\vert F]\vert m]=\mathbb{E} [\frac{9}{4} \mathbb{1}_{F=1}+2 \mathbb{1}_{F=2}\vert m]=\frac{9}{4} \mathbb{1}_{m\in \{1,3\}}+2 \mathbb{1}_{m\in\{2,4\}},$$ what is quite the same as before, so i think i made a mistke, but don't know where.

mastermind
  • 25

1 Answers1

0

"...what is quite the same as before, so I think I made a mistake..."

No, you did not.

Let $X,Y,Z$ be random variables with $\sigma(Y)\subseteq\sigma(Z)$ or equivalently $Y=k\left(Z\right)$ for some function $k$.

Observe that for any suitable function $g$ we have: $$\mathbb{E}\left(\mathbb{E}\left[\mathbb{E}\left[X\mid Z\right]\mid Y\right]g\left(Y\right)\right)=\mathbb{E}\left(\mathbb{E}\left[X\mid Z\right]g\left(Y\right)\right)=\mathbb{E}\left(\mathbb{E}\left[X\mid Z\right]g\left(k\left(Z\right)\right)\right)=$$$$\mathbb{E}\left(Xg\left(k\left(Z\right)\right)\right)=\mathbb{E}\left(Xg\left(Y\right)\right)$$

This allows the conclusion that $\mathbb{E}\left[\mathbb{E}\left[X\mid Z\right]\mid Y\right]=\mathbb{E}\left[X\mid Y\right]$

Setting $X=\text{wage}$, $Y=F$ and $Z=m$ we find: $$\mathbb{E}\left[\mathbb{E}\left[\text{wage}\mid m\right]\mid F\right]=\mathbb{E}\left[\text{wage}\mid F\right]$$

drhab
  • 151,093