Variance of a linear combination if parameters are random instead of fixed (as in bayesian estimation vs MLE)

If X is a r.v. and beta a fixed parameter (say, an MLE estimate), we know that: Var(beta*X) = beta^2 * Var(X).

I wonder whether this holds in a bayesian framework, where beta is itself a r.v.

Thank you

If you assume the random variables are independent I believe there is an explicit formula.

V(XY) = (E[X])^2*V[Y] + (E[Y])^2*V[X] + V[X]V[Y]

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Thank you very much!