i have a problem and i can't figure out any solution.
Suppose i have this game: i throw a die untill i get a 6. Every time i throw the dice i pay -1 and when i get the 6 i win 5. (Nb: when i obtain the first 6 the game end).
I call $X$ a radom variable that is the number of attempts to get a 6. $X$ has a geometric distribution with parameter $\theta = \frac{1}{6}$.
Now define a variable that tells me how much i win/loss: $Y = 5 - X$; this is my transformation.
The problem is that i don't know how to compute the Cumulative distribution function for the $Y$.
I tried: $F_Y(y) = P(Y \le y) = P(5 - X \le y) = P(X \ge y - 5) = 1 - P(X < y-5) = 1 - F_X(y -5 -1)$ sustituting: $F_Y(y) = 1 - (1 - (\frac{5}{6})^{y - 6}) = (\frac{5}{6})^{y - 6}$
That is wrong! Can you help me finding the mistake?
Thank you, bye.