Question

Let 𝑋₁, 𝑋₂, 𝑋₃ be a random sample from an 𝑁(𝜃, 1) distribution, where 𝜃 ∈ ℝ is an unknown parameter. Then, which one of the following conditional expectations does NOT depend on 𝜃 ?

• A. 𝐸(𝑋₁+𝑋₂−𝑋₃ | 𝑋₁ + 𝑋₂)
• B. 𝐸(𝑋₁ + 𝑋₂ − 𝑋₃ | 𝑋₂ + 𝑋₃)
• C. 𝐸(𝑋₁ + 𝑋₂ − 𝑋₃ | 𝑋₁ − 𝑋₃)
• D. 𝐸(𝑋₁ + 𝑋₂ − 𝑋₃ | 𝑋₁ + 𝑋₂ + 𝑋₃)

Answer 1

The Correct Option is D

To determine which conditional expectation does not depend on \(\theta\), we analyze each of the given options. Note that \(X_1, X_2, X_3\) are independent and identically distributed random variables from the normal distribution \(N(\theta, 1)\), having the form:

1. For a normal distribution \(N(\mu, \sigma^2)\), if \(Y = aX + b\), then \(E(Y) = aE(X) + b\). 
2. The expected value \(E(X_i) = \theta\) for each \(X_i\), since they are distributed as \(N(\theta, 1)\).

Now let's evaluate each option:

• Option 1: \(E(X_1 + X_2 - X_3 \mid X_1 + X_2)\)
  • The term \(X_1 + X_2\) is sufficient for the conditional expectation. Hence, this depends on \(\theta\).
• Option 2: \(E(X_1 + X_2 - X_3 \mid X_2 + X_3)\)
  • This can still be influenced by the mean \(\theta\) as it's not independent of other components.
• Option 3: \(E(X_1 + X_2 - X_3 \mid X_1 - X_3)\)
  • Similar to previous reasoning, \(\theta\) impacts the expectation here.
• Option 4: \(E(X_1 + X_2 - X_3 \mid X_1 + X_2 + X_3)\)
  • This condition sums all three random variables together. Given that we are summing over all \(X_i\), the overall mean \(\theta\) cancels out, leading to a result that does not depend on \(\theta\).

Hence, the correct answer is the conditional expectation that does not depend on \(\theta\), which is Option 4: \(E(X_1 + X_2 - X_3 \mid X_1 + X_2 + X_3)\).