Consider two non-independent gaussian random variables:

$$(t,c)βΌtext{BiNormal}((ππ‘,ππ),(ππ‘,ππ),π)$$

I’m interested in understanding when (i.e. for which values of the distribution parameters) it’s true that $$frac{partial E(t |c<f(t)+x)}{partial x}>0$$

where

```
f(t_):=b*CDF(NormalDistribution(ΞΌa, Οa), t)
```

**What I’ve tried:**

- I’ve tried generating an analytical expression for the conditional expectation, but this fails (output same as input)

`Expectation(t | c < b*CDF(NormalDistribution(ΞΌa, Οa), t) + x, {t,c}~BiNormal((ππ‘,ππ),(ππ‘,ππ),π))`

- I’ve also tried spelling out the expectation via integrals over the joint distribution, and then taking derivatives. This does produce a valid analytical output, but I cannot reduce the expression (error message
*“This system cannot be solved with the methods available to Reduce”*)

`D(Integrate(t*PDF(BinormalDistribution({Β΅t,Β΅c},{Οt,Οc},Ο),{t,c}),{c,-β,CDF(NormalDistribution(Β΅a,Οa),t)+x},{t,-β,β)})/ Integrate(PDF(BinormalDistribution({Β΅t,Β΅c},{Οt,Οc},Ο),{t,c}),{c,-β,CDF(NormalDistribution(Β΅a,Οa),t)+x},{t,-β,β)}),x))`

Here is a related question on the analytical treatment of this problem: https://math.stackexchange.com/questions/3692958/parameters-for-which-conditional-expectation-of-non-independent-gaussian-variabl