Independence of $X$ and $2X$ Are these two random variables independent? Unfortunately, I don't know probability theory enough to answer this question. I know for a fact that if $X$ and $Y$ are independent random variables and $g$,$h$ are measurable functions, then $g(X)$ and $h(Y)$ are independent as well. However, I don't know if that can be used here.
I ran into this question in my work, where someone asked if the following moment generating function was valid:
$$M_{X}(t)M_{X}(2t) $$
and stated that this was, indeed, a valid moment generating function for a random variable $X+Y$ where $Y$ follows the same distribution of $2X$.
But is it true that you can write the moment generating function of $X+Y$ as such? (Basically, I doubt that $X$ and $2X$ are independent.)
My main question: Does there exist a random variable $X$ such that you can write the moment generating function of $X + 2X$ as stated above (assuming the MGFs exist)?

The formula for the mgf of a sum $X+Y$ is correct when $X$ and $Y$ are independent.
Apart from  a few degenerate examples, $X$ and $2X$ are not independent. 
For instance, toss a fair coin, and let $X=1$ if we get a head, and $X=0$ otherwise. 
Then $\Pr(X=1\cap 2X=0)=0$. But $\Pr(X=1)\nee 0$, and $\Pr(2X=0)\ne 0$, so 
$$\Pr(X=1\cap 2X=0)\ne \Pr(X=1)\Pr(2X=0).$$
The above was a formal showing of non-independence. At the more informal level, if we know that $X=1$, we know a great deal, indeed everything, about $2X$. 
Edit: For the question about whether the mgf of $X+2X$ is ever the product of the mgf, let $X=0$ with probability $1$. 
