Analytical expression for distribution of a function of gaussian random variables

Question

Suppose I have two non-independent gaussian random variables:

$(T,C)\sim \text{BiNormal}[(\mu_t,\mu_c),(\sigma_t,\sigma_c),\rho]$

Let's also say the function $F(.)$ is the CDF of a Normal distribution with mean $\mu$ and variance $\sigma$.

I would like to have an expression for $Pr[C<F(T)]$.

For this I can of course define $Z=C-F(T)$, and so the relevant question becomes: Can I derive an expression for the distribution of $Z$?

PS. I'm interested specifically in an expression for $C\in(0,1)$.

Answer 1

Here is a numerical approach using Mathematica:

(* Define bivariate normal distribution *)
dTC = BinormalDistribution[{μT, μC}, {σT, σC}, ρ];

(* Parameters of interest *)
parms = Thread[{μT, μC, σT, σC, ρ, μ, σ} -> {0, 1, 1, 2, 1/2, 1, 1}]
(* {μT -> 0,μC -> 1,σT -> 1,σC -> , ρ -> 1/2,μ -> 1,σ -> 1} *)

NProbability[c < CDF[NormalDistribution[μ, σ], t], {t, c} \[Distributed] dTC] /. parms
(* 0.34283 *)

For the distribution of $Z$, random samples can be used to construct a nonparametric density estimate:

n = 10000;
tc = RandomVariate[dTC /. parms, n];
z = #[[2]] - CDF[NormalDistribution[\[Mu], \[Sigma]] /. parms, #[[1]]] & /@ tc;
skd = SmoothKernelDistribution[z];
Plot[PDF[skd, z], {z, -10, 10}]

R could also be used to provide numerical estimates.

Answer 2

generic answer is $\int_0^1 dC \int dT p(C,T) \Theta (F(T)-C) $ where $\Theta$ is step function of zero for argument less than zero and one for argument greater than zero