Let be $Y$ Beta$(\alpha, \beta )$ distributed random variable. Further more let $X$ conditional on $\{Y=y \}$ geometric distributed with paramter $y$. Determine the distribution of X.
I know $Y$ is defined by the distribution function $$f_y=\frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}x^{\alpha-1}(1-x)^{\beta-1}$$ Further more I think $X$ conditional on $\{Y=y \}$ is given by $$f_{(x|y)}= 1-(1-y)^n$$ According to wikipedia $$f_{(x|y)}=\frac{f(x,y)}{f_y}$$ So $f_y \cdot f_{(x|y)}=f(x,y)$. This does not seem to be simplified easily. According to this one wants $f_x=f_{(y|x)} \cdot f(x,y)$. But one is not able to calculate $f_{(y|x)}$. Or what am I missing?