It will be much better if you initialize mussigma inside the init function and return it to the user.
Then pass mussigma to the next function that will use it as a parameter.
Something like:
module Mixture
export init, compute
function init(...)
mussigma = Vector{...}(undef,...)
...
return mussigma
end
function compute(mussigma)
...
return result
end
end
using Mixture
mussigma = init(...)
result = compute(mussigma)
Try to forget completely about the possibility of using global variables, that is the best advice.