Consider two random variables, $X$ and $Y$, with the following properties: $X\mid Y\sim N(Y,s^2)$ and $Y\sim N(\mu,\sigma^2)$.
Does $X\mid Y>y$ follow a normal distribution as well? If so, what are its parameters and how can I prove this? If not, can we say anything interesting about the distribution of $X\mid Y>y$ beyond giving it as a complicated integral?
The complicated integral being: $\Pr(X<x\mid Y>y) = \int_{-\infty}^x \int_y^\infty f_{X\mid u}(t)f_Y(u)\, du\, dt$, where $f_Y$ is the density function of a $N(\mu,\sigma^2)$ and $f_{X\mid u}$ the density function of a $N(u,s^2)$ distribution.
Since the unconditional distribution of $X$ is $N(\mu,\sigma^2 + s^2)$, and one way to find this is by taking the above integral and replacing $y$ by $-\infty$, my intuition was that for $X\mid Y>y$ you would get something like a "chopped off" normal distribution, i.e., the answer would be no.
But my colleague's intuition was that, since sums of normal distributions are normal, and an integral is like a weighted sum with uncountably many terms, the result would be normal.
I'd be very grateful if someone could settle the question.