How to transition from pure biology PhD to bioinformatics/biostatistics?
As a PhD student in Math with Computer Science background who is working in Bioinformatics exclusively: it is definitely possible for you! 

In the Computational Biology Institute at The George Washington University where I work as GRA we have PhD students, Post-docs as well as Faculty from different backgrounds (including pure biology) working purely in bioinformatics area.

I do believe you don't have to switch as much, as you think. It only takes a group that is more focused on the "informatics" part more, than on biology. But still caring about Biology to change the main daily routine of yours. And your background will be needed, as bioinformatics can not exists without biological reasoning and biological guidance / insights. 

If I were you, my first step would be to find what is the most "informatics" related problem close to your current research. It might be not your group, who is doing it, but still. Educate yourself a bit about such research by reading related articles. And after that it is as simple as writing a couple of emails to PIs of respective studies and telling them, that you are very interested in collaboration on their topic. This might yield a joint publication or even a post-doc position offer from respective lad.

I wouldn't count much on pure "career" in statistics, as it usually requires more math background. But biologists in bionformatics, who at least knows how to do statistics are very hard to find. So the more you know from the programming / statistics side, the better you are off go to more informatics groups.