Assume that $v_s \sim N(\mu_s,\sigma_s^2)$ and $v_b \sim N(\mu_b,\sigma_b^2)$, denote their correlation by $\rho$, and assume they are jointly normally distributed. How would I assess $E[v_b|v_s\leq c]$ where $c$ is some constant?

I figured it would be of the form $$E[v_b|v_s\leq c]=\frac{1}{P(v_s\leq c)}\int_{-\infty}^cE[v_b|v_s]f_{v_s}dv_s$$ however I'm not sure this is right. Not sure how to conceptualize the approach here. Thanks!

  • $\begingroup$ You can't evaluate this expectation unless you make a more specific assumption about the joint distribution: most likely you intend for it to be Binormal. In that case your question is answered at stats.stackexchange.com/questions/356023 (because having nonzero means is a trivial complication). Otherwise, what do you want to assume about the joint distribution? $\endgroup$ – whuber Nov 12 '20 at 20:16
  • $\begingroup$ A possible approximate solution via a numerical method? Start with the associated conditional Normal given a point in the interval, which is well known (a source of the Expected value and height of the PDF at that point). Average the Probability X Expected values over a grid of points in the interval of interest. Apply a probability normalization factor, given that the value is in the cited interval. $\endgroup$ – AJKOER Nov 12 '20 at 20:53