# Condition Random Variable on Range of Another Random Variable [duplicate]

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!

• 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? – whuber Nov 12 '20 at 20:16
• 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. – AJKOER Nov 12 '20 at 20:53