Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I was thinking what is the $\sigma$-algebra generated by the random variable $Z= \mathbb{I}(X+Y=0)$ where $X,Y\sim\operatorname{Bern}(p)$ iid. (Note: $\mathbb{I}$ is the indicator function.)

share|cite|improve this question
up vote 2 down vote accepted

It's the smallest $\sigma$-algebra making $Z$ measurable. As the values of $Z$ are $0$ or $1$, $\sigma(Z)=\left\{\emptyset,\Omega, Z^{-1}\left(\{0\}\right),Z^{-1}(\{1\})\right\}$ and $$Z^{-1}(\{0\})=\{X\neq -Y\}=\{X=1,Y=1\}\cup\{X=-1,Y=-1\}.$$

In order to compute $\mathbb E\left[X\mid Z\right]$, we compute $\mathbb E\left[X\mathbf 1\{Z=0\}\right]$ and $\mathbb E\left[X\mathbf 1\{Z=1\}\right]$. We have $$\mathbb E\left[X\mathbf 1\{Z=0\}\right]=\mathbb E\left[X\mathbf 1\{X=1\}\mathbf 1\{Y=1\}\right]+\mathbb E\left[X\mathbf 1\{X=-1\}\mathbf 1\{Y=-1\}\right]=p^2-(1-p)^2,$$ and we compute similarly $\mathbb E\left[X\mathbf 1\{Z=1\}\right]$. Then we use the formula which gives the conditional expectation of a random variable with respect to a $\sigma$-algebra generated by a finite partition.

share|cite|improve this answer
Thanks Davide! (btw you have made a typo with the minus signs) Could you also tell me what the conditional expectation of X given Z is? – adamG Oct 29 '12 at 23:28

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.