Question
Statistics Question on Multivariate Distributions
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
𝐸(𝑋1 + 𝑋2 − 𝑋3 | 𝑋1 + 𝑋2 + 𝑋3)
Explanation
Solution
The correct option is (D): 𝐸(𝑋1 + 𝑋2 − 𝑋3 | 𝑋1 + 𝑋2 + 𝑋3)