i. If Y is the number of turns in the game, this is also the number of rounds until the red disc comes up for the first time. I claim that and its mass function is , y=1,2,... ie the number of trials until the first success (success=red disc comes up)
For Mx(t) to be well-defined, , and likewise, denominator in My(t) should be strictly positive, ie
(thinking to myself, since 1-p<1, ln(1-p)<0 so t must be positive...)
iii. Z is the function of Y and X, which, in turn, are pairwise independent. Therefore
by the way, I am not sure if I can use t parameter in all three functions, doesn't sound right, as they are three differnet mgfs? But if I use three different letters for parameters, I can never multiply
Anyways, I get
and since I cannot, cannot, cannot express this as a polynomial (must be something to do with expension of e^t?? but I got messy in the end and gave up) I took the first derivative of the fraction and found E(Z) as 1st derivative of Mz(t) around t=0. The calculation was a bit tedious so I'll just say that I got
And since the direct calculation of the 2nd derivative from the fraction would be even more tedious, I welcome any pointers on how to express Mz(t) as polynomial so to do the Var(Z) in a proper way.