Final answer:
The specified parameters for μ and τ regarding their means and variances are not aligned with the properties of a normal-gamma distribution, thus two random variables cannot jointly have a normal-gamma distribution with these parameters.
Step-by-step explanation:
To show that two random variables μ and τ cannot have a joint normal-gamma distribution with the given parameters, we must understand that a normal-gamma distribution is a type of compound distribution where the normal distribution's variance is itself randomly distributed according to a gamma distribution. However, from the properties stated, μ has E(μ) = 0 and Var(μ) = 1, which are consistent with a normal distribution, but τ's expectation E(τ) = 1/2 and variance Var(τ) = 1/4 do not align with a gamma distribution's properties. The gamma distribution's variance is typically a function of its shape and scale parameters and cannot be arbitrarily chosen to fit the values given without affecting the remaining distribution parameters.
Additionally, the query appears to mix up various theoretical statistics and probability concepts that do not pertain directly to the question of establishing a joint normal-gamma distribution for the random variables μ and τ. Specifically, the reference to the empirical rule and central limit theorem relates to a normal distribution's properties and not to the demonstration of a joint normal-gamma distribution existing with specified parameters.