# ----------------------------------------------------------------
# The covariance must be positive semi-definite: exactly what that means is not so easy,
# but at least all values must be positive, and further the off-diagonal values must not
# be too high.
# ----------------------------------------------------------------


# ----------------------------------------------------------------
# Yes, any "straight cut" through it will be normal distributed. Think about that any
# linear transformation is also normal distributed, and a linear projection is a linear
# transformation into a subspace, e.g.\ R^2 -> R, hence it will also be normal distributed.
# ----------------------------------------------------------------


# ----------------------------------------------------------------
# First part to be filled in:
mu2 <- ??
Sigma2 <- ??

# Can be:
mu2 <- A %*% mu + b
Sigma2 <- A %*% Sigma %*% t(A)
# ----------------------------------------------------------------


# ----------------------------------------------------------------
# Second part to be filled in:
# First the mean
??
# Then the variance, it's two not independent variables
??

# Can be:
# First the mean
t(mu) %*% theta
# Then the variance, it's two not independent variables
# As matrices
t(theta) %*% Sigma %*% theta
# Written out for each variable
theta[1]^2 * Sigma[1,1] + theta[2]^2 * Sigma[2,2] + 2 * theta[1] * theta[2] * Sigma[1,2]
# We also must remember the variance from the independent error, then just add it
t(theta) %*% Sigma %*% theta + sigma^2
# ----------------------------------------------------------------