A continuation of this question
"Suppose there are two independent random variables: $\mathbb{A}$ ~$\mathit{N}(0,1)$ [ie standard normal] and $\mathbb{B}$ that is ~Bernoulli with $P(\mathbb{B} = 1) = P(\mathbb{B} = -1) = 0.5$."
From the answer: $\mathbb{C}$ ~$\mathit{N}(0,1)$ where $\mathbb{C} = \mathbb{AB}$
So, now is there a simple way to check the following for $\mathbb{A}$ and $\mathbb{C}$?
Independence: $$f_{\mathbb{AC}}(a,c) = f_{\mathbb{A}}(a)f_{\mathbb{C}}(c)$$
Uncorrelated: $$\text{E}[\mathbb{AC}] = \text{E}[\mathbb{A}]\text{E}[\mathbb{C}]$$
In order to find $f_{\mathbb{AC}}(a,c)$ do you have to apply Bayes' Rule and first find the conditional? Or is there a trick since $\mathbb{A}$ and $\mathbb{C}$ are Gaussian?
They're uncorrelated as $\Bbb{AC}=\Bbb{A}^2\Bbb{B},\,\Bbb A$ each have zero mean, but they're not independent since each value of $\Bbb A$ is consistent with a different pair of values of $\Bbb C$. Wikipedia has discussed this.