Suppose $Y\sim N(0,X^2)$, and $X\sim Bernoulli(0.5)$ on $\{1,2\}$, how to prove (or rigorously argue) that $X,Y$ are not independent Suppose $Y\sim N(0,X^2)$, and $X\sim Bernoulli(0.5)$ on $\{1,2\}$, how to prove (or rigorously argue) that $X$ and $Y$ are not independent?
Can I proceed as follows: try to show that there exists $x$ and $y$ such that $Pr(Y\leq y|x)\neq Pr(Y\leq y)$?
This could be shown as follows:
$Pr(Y\leq y)=0.5\Phi(y)+0.5\Phi(\frac{y}{2})$,
$Pr(Y\leq y|X=1)=\Phi(y)$
Thus $Pr(Y\leq y)-Pr(Y\leq y|X=1)=0.5[\Phi(\frac{y}{2})-\Phi(y)]\neq0$ for $y\neq 0$.

$P(|Y| > 1.96\, |\, X = 1) < 0.06,$ but $P(|Y| > 1.96,|\, X = 2) > .32.$ This can be shown using printed normal tables. Illustration by simulation in R below:
set.seed(722)
x = sample(1:2, 10^6, rep=T)
y = rnorm(10^6, 0, x)
mean(abs(y[x==1]) > 1.96)
[1] 0.05046732
mean(abs(y[x==2]) > 1.96)
[1] 0.3280174

