MGF are very confusing

I can't ask all my questions about them, so I'll try to ask just one:



M[X+Y](T) = .09e^-2t + .24e^-t + .34 + .24e^t + .09e^2t



X and Y are IID, and we want to find Pr(X<=0)



Ok, so it says that M[x](t)=M[y](t), which is the first confusing part. Why are two MGFs the same just because the variables are IID?



It goes on to say that because of this, M[X+Y](T) = M[X](T)*M[X](T) = M[X](T)^2, sure. So we want to write the above as a square of another expression, which makes sense. Some fancy mathwork gets the answer of M[X](T) = .3e^t + .4 + .3e^-t --> X = {-1,0,1} with Pr {.3,.4,.3} and the answer is .7



This took about 10 minutes to do the math on and was very confusing, so it offers a second solution:



You can recognize X + Y = {-2,-1,0,1,2} with Pr {.09, .24, .34, .24, .09}, which gives you:



-2 = (-1) + (-1)

-1 = (-1) + (0)

-1 = (0) + (-1)

0 = (0) + (0)

0 = (-1) + (1)

0 = (1) + (-1)

1 = (1) + (0)

1 = (0) + (1)

2 = (1) + (1)



Which apparently leads to X = {-1,0,1} with Pr {.3,.4,.3} again



How in the world do you get from the second to last step to the last step? There are 9 possibilities, so dividing by 9 doesn't give you a nice .3 or .4. If you multiply things by factors of .09 and .34 etc. you don't get a nice .3 or .4. And why aren't we also adding these to the above:



- 2 = (-2) + (0)

- 2 = (0) + (-2)

2 = (2) + (0)

2 = (0) + (2)



:squintyeyes:

MGF are very confusing