Statistical Machine Learning Question

Consider a categorical-Gaussian mixture model, that is, a joint random
variable X, Y with categorical random variable X which has K = 2 possible
outcomes, X ∈ {1, 2}, with parameter vector p with entries p1, p2. The Gaussian
random variable Y has dimension D = 1, with mean parameters µ1, µ2 and
standard deviations σ1, σ2.

Find an explicit expression for the joint probability density of X, Y .