Under X|θ ~ N(θ, σ^2) and θ ~ N(μ, τ^2), the marginal distribution of X is Normal with mean μ and variance σ^2 + τ^2.

Under X|θ ~ N(θ, σ^2) and θ ~ N(μ, τ^2), the marginal distribution of X is Normal with mean μ and variance σ^2 + τ^2.

• A. True
• B. False
• C. Only if σ^2=τ^2
• D. Cannot be determined

Answer: (A)

The distribution of X remains Normal because you’re adding two Gaussian pieces: θ, which shifts the mean of X, and an independent Gaussian noise with variance σ^2. In other words, X can be viewed as θ plus a random error Z ~ N(0, σ^2). Since θ ~ N(μ, τ^2) and Z is independent of θ, the sum X = θ + Z is the sum of two independent Normals, which is again Normal with mean equal to the sum of the means and variance equal to the sum of the variances. Therefore, the unconditional distribution of X is Normal with mean μ and variance σ^2 + τ^2. This confirms the statement as true, and it does not require σ^2 to equal τ^2. The result also follows from the law of total variance: E[X] = μ and Var(X) = Var(θ) + Var(Z) = τ^2 + σ^2.

The distribution of X remains Normal because you’re adding two Gaussian pieces: θ, which shifts the mean of X, and an independent Gaussian noise with variance σ^2. In other words, X can be viewed as θ plus a random error Z ~ N(0, σ^2). Since θ ~ N(μ, τ^2) and Z is independent of θ, the sum X = θ + Z is the sum of two independent Normals, which is again Normal with mean equal to the sum of the means and variance equal to the sum of the variances. Therefore, the unconditional distribution of X is Normal with mean μ and variance σ^2 + τ^2. This confirms the statement as true, and it does not require σ^2 to equal τ^2. The result also follows from the law of total variance: E[X] = μ and Var(X) = Var(θ) + Var(Z) = τ^2 + σ^2.