When comparing MGFs why is $X+2Y \ne X+Y+Y$?

As the title states. I have $X \sim ChiSquare(2)$ and $Y \sim ChiSquare(2)$, i.e. they are both distributed as chi-squares with 2 degrees of freedom and are independent.

Define a new random variable $U = X+2Y$. After some calculation I have found that $X+2Y \ne X + Y + Y$. What is the reasoning for this?

EDIT: For clarity, my thinking was that if $U=X+2Y$ we could write it as $U=X+Y+Y$, as $2Y=Y+Y$.

EDIT: I attempted to solve this by making use of the fact that the MGF of the sum of independent random variables, is the product of the indiviudal MGFs.

Here for $X+Y+Y$ I treated this as $M_X(t) * M_Y(t) *M_Y(t)$, but this supposedley leads to the incorrect answer.

Solution 1:

When you write $X + Y + Y$, what exactly do you mean by $Y + Y$? Does this represent the sum of two independent random variables, each of which follows a chi-square distribution with $2$ degrees of freedom, or does it represent the sum of a single chi-squared random variable with itself? In the former case, the notation is incorrect and would not be acceptable: it should read $Y_1 + Y_2$, where $Y_i \sim \operatorname{ChiSquare}(2)$ for $i \in \{1, 2, \ldots\}$.

To illustrate, suppose $Y$ is such a chi-squared random variable. Using a computer, I generated the realization $Y = 1.36952$. Now, when I write $Y + Y$, what I mean is the realization $Y + Y = 2.73905$. However, if what is intended is the sum of two independent and identically distributed chi-squared random variables, then you could have $$Y_1 + Y_2 = 1.36952 + 2.37966 = 3.74918.$$ The second realization, although coming from the same distribution, has a different value than the first.

Whenever something like $Y+Y$ is written, even for random variables, it should be clear that $Y+Y = 2Y$ identically. This is why, if the intent was to represent two IID chi-squared random variables, rather than twice a single chi-squared random variable, such notation is incorrect, and the notation $Y_1 + Y_2$ or something similar to distinguish the two variables, must be used.

After seeing your edit, the error is that if $U = X + Y + Y$, you cannot write $$M_U(t) = M_X(t)M_Y(t)M_Y(t).$$ This is incorrect because the two instances of $Y$ are always equal, hence not independent. Then the theorem does not apply. If $U = X + Y_1 + Y_2$ where each variable in the sum is independent, then you could write $M_U(t) = M_X(t) M_{Y_1}(t) M_{Y_2}(t)$.