Take samples $X_i$ from normal distribution $N(0, 1)$, find constant $C$ so that $C[(X_1+X_2+X_3)^2+(X_4+X_5+X_6)^2]\sim \chi^2$

Question

Take samples $X_1$, $X_2$, $X_3$, $X_4$, $X_5$, $X_6$ from normal distribution $N(0, 1)$, $Y=(X_1+X_2+X_3)^2+(X_4+X_5+X_6)^2$. Find constant $C$ so that $CY\sim \chi^2$.

The answer is 1/3. I don't have a clue to this question.

Answer 1

Since $X_1 + X_2 + X_3 \sim N(0, 3)$ and $X_4 + X_5 + X_6 \sim N(0, 3)$ are independent, we have that \begin{align*} \frac{X_1 + X_2 + X_3}{\sqrt{3}} \sim N(0, 1) \quad \text{and} \quad \frac{X_4 + X_5 + X_6}{\sqrt{3}} \sim N(0, 1) \end{align*} and so \begin{align*} \frac{(X_1 + X_2 + X_3)^2}{3} \sim\chi^2_1 \quad \text{and} \quad \frac{(X_4 + X_5 + X_6)^2}{3}\sim\chi^2_1 \end{align*} And so \begin{align*} \frac{1}{3}\left((X_1 + X_2 + X_3)^2 + (X_4 + X_5 + X_6)^2\right) \sim \chi^2_2 \end{align*}