Let $X$ and $Y$ be i.i.d. $N(0, 1)$, and let $S$ be a random sign (1 or -1, with equal probabilities) independent of $(X, Y)$.
Determine whether or not $(SX, SY)$ is Multivariate Normal.
My attempt is:
$k_1SX + k_2SY = S(k_1X + k_2Y)$ for some constants $k_1$ and $k_2$
As $W = (k_1X + k_2Y)$ is normally distributed, we can show that $Z = SW$ is normally distributed as,
\begin{align*} P(Z \leq z) & = P(Z\leq z|S=1)P(S=1) + P(Z\leq z|S=-1)P(S=-1)\\ & = 0.5P(W \leq z)+0.5P(W\leq z)\\ & = P(W \leq z). \end{align*}
Hence the distribution of $Z$ is normal for any choice of $k_1$ and $k_2$.
Is there anything wrong in this proof.
Attempt 2: As for the calculations given in Let X and Y be i.i.d. N(0, 1), and let S be a random sign (1 or -1, with equal probabilities) independent of (X, Y).
\begin{align*} P((SX,SY)∈B)&=P((X,Y)∈B,S=1)+P((−X,−Y)∈B,S=−1) \\ &= P((X,Y)∈B)P(S=1)+P((−X,−Y)∈B)P(S=−1)\\ &= 0.5*P((X,Y)∈B) + 0.5*P((−X,−Y)∈B)\\ &= P((X,Y)∈B) \end{align*}
Which one is the correct answer?