Question

Let \(X_1,X_2,X_3\) be three independent random variables such that \(X_1\sim N\left(0,\frac12\right)\), \(X_2\sim N(0,2)\), and \(X_3\sim N(0,4)\). Consider the following statements.

• A. \(P\) is correct and \(Q\) is NOT correct
• B. \(P\) is NOT correct and \(Q\) is correct
• C. Both \(P\) and \(Q\) are correct
• D. Neither \(P\) nor \(Q\) is correct

Hint:

An \(F\)-distribution is formed by the ratio of two independent chi-square variables divided by their degrees of freedom, and the ratio of two independent standard normal variables follows a standard Cauchy distribution.

Answer 1

The Correct Option is C

Step 1: Turn the normals into standard ones.
Write $Z_1=\sqrt2 X_1$, $Z_2=X_2/\sqrt2$, $Z_3=X_3/2$, all independent $N(0,1)$.

Step 2: Rewrite $P$'s ratio.
$\dfrac{16X_1^2}{2X_2^2+X_3^2}$ becomes $\dfrac{2Z_1^2}{Z_2^2+Z_3^2}=\dfrac{Z_1^2/1}{(Z_2^2+Z_3^2)/2}$, an $F_{1,2}$. So $P$ is correct.

Step 3: Set up $Q$.
Let $U=2X_1-X_2$, $V=2X_1+X_2$. Both are normal with variance $4$ and $\mathrm{Cov}(U,V)=\mathrm{Var}(2X_1)-\mathrm{Var}(X_2)=2-2=0$.

Step 4: Identify the ratio.
Jointly normal and uncorrelated means independent, so $U/V$ is the ratio of two independent mean zero normals of equal spread, which is standard Cauchy. $Q$ is correct.

Step 5: Conclude.
Both hold, option (C).
\[ \boxed{(C)} \]