A friend asked me about a (homework) question of the following form:
The random vector $(X, Y)$ is bivariate normal with means $=0$ and standard deviations $=1$, the correlation is $\rho = \frac{1}{2}$. Find $\mathbb{P}(Y > 0 \mid X = 1)$.
So I've never taken calculus-based probability, so my first instinct is to say that the answer is 0, because we're conditioning on an event of probability 0. But I feel like this is a cop-out answer, so my intuition tells me to just pretend like we're working in one dimension and do something like:
$$\mathbb{P}(Y > 0 \mid X = 1) = \frac{\int\limits_{0}^\infty f_{XY}(1,y) \; dy}{\int\limits_{-\infty}^\infty f_{XY}(1,y) \; dy} $$
where $f_{XY}$ is their joint distribution.
So I have two questions. First, is this the correct interpretation? Second: if it is, it seems to me like finding a closed-form answer is unreasonable for a calculus-based probability class, since the integrand is an exponential with a nontrivially quadratic argument, but maybe I'm missing something.
In general, evaluating this integral would be too difficult for a probability class, but you don't need to evaluate the integrals thanks to a lot of special properties of multivariate normal distributions. Take a look at Wiki: Multivariate normal distribution for an overview.