Let U be (continuous) uniform on (0,1) and 0 < p < 1. Show that X = ceiling( ln(U)/ln(1-p)), has geometric distribution.
I understand that I need to start with P(U =< x) and I will need to integrate but I am not sure how to apply this to this particular question.
For positive integer $k$ we find:
$$P\left(\lceil\ln\left(U\right)/\ln\left(1-p\right)\rceil>k\right)=P\left(\ln\left(U\right)/\ln\left(1-p\right)>k\right)=P\left(U<\left(1-p\right)^{k}\right)=\left(1-p\right)^{k}$$