Let $X_1\sim Laplace(0,\sqrt{1/2})$ and $X_2 \sim Laplace(1/2,\sqrt{1/2})$. Are $X_1$ and $X_2$ independent?

Question

Let $X_1\sim Laplace(0,\sqrt{1/2})$ and $X_2 \sim Laplace(1/2,\sqrt{1/2})$. Are $X_1$ and $X_2$ independent?

I understand that in case of independence, the joint pdf is the product of the marginal pdfs. I know how to find marginals from joint pdf but I'm not sure how to find $f_{X_1,X_2}$. Also, can I use the expectation to prove independence?

Answer 1

People in the comments section are trying to explain that we really don't have enough information. Maybe this convinces you...

I construct $X_1$ and $X_2$, with the same distributions you have in the question, which are dependent in one case and independent in another. Let $Y\sim \text{Laplace}(0,\sqrt{1/2})$. I give you two situations.

Dependent. Define $X_1 = Y$ and $X_2=Y+\frac 1 2$. Now we have that $X_1$ and $X_2$ are dependent, you see why?

Independent. Let $Y_1$ and $Y_2$ be independent versions of $Y$. Now define $X_1=Y_1$ and $X_2=Y_2+\frac 1 2$. In this case they are independent, you see why?

So, we really can't tell anything about dependence if you just tell us the marginal distributions.