Let $v\sim\text{Normal}\left(\mu,\sigma_v^2\right)$ a random variable with $\mu>0$ and $u\sim\text{Normal}\left(0,\sigma_u^2\right)$

Let $k\sim\text{Binomial}\left(N,p\right)$ a random variable with $N\geq 2$ an integer and $p\in\left[0,1\right]$.

Finally, let $q= k\times\beta\times\left(v-\mu\right)+u$.

What is $\mathbb{E}\left(v\mid q\right)$?

I computed $\mathbb{E} q = 0$, $\text{var} \left(q\right) = \sigma^2_u + \beta^2 \sigma^2_v \times \left[p \left(N(1-p) + N^2 p\right)\right]$, and $\text{covar}\left(q,v\right)=\beta \sigma^2_v N p$ (and verified the results numerically).

I don't think one can use in this case the following regression (??): $$ \mathbb{E}\left(v\mid q\right) = \mu+\frac{\text{covar}\left(q,v\right)}{\text{var} \left(q\right)} q, $$ since $q$ is essentially a mixture of normals ($k$ is random).


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.