How to calculate the sum or difference of two probability generating functions? Suppose I have probability generating functions $$G_{X}(t) = 0.1t+0.2t^{2}+0.7t^{3}\quad\text{and}\quad G_{Y}(t)=0.5+0.4t^{2}+0.1t^{3}.$$ In other words, the random variable $X$ gets the discrete values $P(X=1)=0.1, P(X=2)=0.2,\ \text{and}\ P(X=3)=0.7$.
How do I calculate the sum or difference of $G_{X}$ and $G_{Y}$? It would feel intuitive to write $$G_{X}(t)+G_{Y}(t) = 0.5 + 0.1t + 0.6t^{2}+0.8t^{3}$$
but the coefficients sum over $1$. If I divide each coefficient by $2$ then they sum again to $1$. Is that the correct to way to approach this?

Edit: Sorry for being unclear. I was trying to ask about $G_{X+Y}$, so the PGF of the sum of $X$ and $Y$. The variables can be assumed to be independent.
 A: $G_{X}(t) + G_{Y}(t)$ is not a probability generating function in general.  But if $X$ and $Y$ are independent you have the case that $G_{X+Y}(t) = G_{X}(t) G_{Y}(t)$, which might be what you're interested in.
A: Are both your probability generating functions independent from each other ? 
The joint probability of two probability generating functions is their product only if the two are independent. If there is some covariance between the two functions, this needs to be taken into account following Bayes' rule:
P(A given B)=P(A)*P(B given A)/P(B)


*

*For independent events: P(A and B) = P(A) *P(B) (because P(B given A)=P(B))

*For dependent events: P(A and B) = P(A) *P(B given A)

A: Based on comments from the OP the question is really asking for product of the two generating functions which is equivalent to asking about the distribution of $Z=X+Y$ where $X$ and $Y$ are independent. Taking the product of the two functions in $t$ and collecting terms gives the answer 
$0.05t+0.1t^2+0.39t^3+0.09t^4+0.3t^5+0.07t^6$
Where the exponent in $t$ tells us the value of $Z$ and the coefficient in front of the $t$ power tells us the corresponding probability. For example, 
$\Pr(Z=1)=0.05$ $\Pr(Z=2)=0.1$ etc. Note this only works if the two variables are independent. If they are dependent one way is to get the answer is to specify the joint distribution and calculate the convolution integral.  
