The question is detailed below:
Construct an example in each of the following settings of three random variables $\alpha, \beta$ and $X$ such that $\alpha$ and $\beta$ are marginally independent but they are no longer conditionally independent given $X$.
(a) All three variables are continuous-valued.
(b) $\alpha$ and $\beta$ are continuous-valued and $X$ is binary valued.
(c) $\alpha$ is binary valued, while $\beta$ and $X$ are continuous.
(d) $\alpha$ and $X$ are binary valued, while $\beta$ is continuous.
In the end, you need to check your answer with demonstrating the product of marginal conditional probability is not equal to the joint conditional probability (e.g. $p(\alpha, \beta \vert X)\ne p(\alpha \vert X)\cdot p(\beta \vert X) $)
I was confused by how to check in the last step for 4 different scenarios. (I do not even know how to write such explicit conditional probability at all)
Note that I have already constructed 4 scenarios as follows (not sure whether correct)
(a) Let these three variables be normal variable with following setting $$\alpha \sim \mathcal{N}(0,1)$$ $$\beta\sim \mathcal{N}(0,1)$$ $$X=\alpha-\beta\sim \mathcal{N}(0,2)$$ (b) $$\alpha \sim \mathcal{N}(0,1)$$ $$\beta\sim \mathcal{N}(0,1)$$ $$ X= \begin{cases} 0& \alpha>\beta\\ 1& \alpha\le\beta \end{cases} $$ (c) $$\alpha \sim p(\alpha)= \begin{cases} \frac{1}{2}& \alpha=1\\ \frac{1}{2}& \alpha=-1 \end{cases}$$
$$\beta\sim \text{Uniform}(0,1)$$ $$ X=\alpha\beta $$ (d) No idea till now.