bernoulli_babe

Any successful PhD student has likely struggled at several points in their training. My first year (Casella-Berger year), I barely managed a 3.5 while my peers were getting Mostly A's and A-'s (for the record, it's pretty hard to score below a B in grad school). After the first year, I was getting better grades than many of my peers who crushed me in the first year. The point of the PhD training is to tax you mentally so that you can start to mature mathematically.
I personally do not think grades or how well you do on the qualifying exam will make you a good researcher. It may be different in some old school professors' eyes, but I think most people these days view the qual / courses as a means to an end. At some point in your career as a graduate student, things will start to click together. And it's very possible you'll never see/use measure theory stuff ever again after taking it.
One of my peers was a bio major in undergrad, and ended up receiving the highest score on the theory portion of our doctoral exam. They had little/no previous exposure to real analysis. They are an extremely hard worker, so there's that. But all this to say, I think it's extremely possible for a bio / CS major to be successful in a statistics / biostatistics PhD program, albeit maybe the latter more than the former.

Contrary to popular belief, I feel that 1st classes at my stats department uses very minimal real analysis. The prerequisite for almost any class is just linear algebra and calculus. You can literally know zero real analysis and do pretty well.
But a level of mathematical maturity is always assumed. It is mostly about problem solving rather than actual knowledge.
A CS major, if solidly done, should have absolutely no problem. A biology major will be more challenging (I'm not talking about "biologists" who are actually theoretical mathematicians or computer scientists in disguise). 
 
 

As someone in industry, an MS is more than enough to get a job as a data scientist or biostatistician. If you're still hesitant about doing research then I highly recommend you go the MS route first then decide after working whether you want to pursue a PhD. 
I personally don't think a PhD is worth it for industry career growth. 
 
EDIT: 
I'd like to add that if you're looking for a PhD program than I'd look into theoretical MS Stats programs. If you're looking for industry then go to well known CS programs that have MS Data Science. Tech companies will be recruiting from there. From what my friends have experienced, the MS Data Science programs are a lot more applied than most MS stats programs. 

Well, depends... if you're a domestic student, then you might be able to get one of those jobs without a PhD -- and sometimes with only a Bachelor's. I have a friend who has BS in Biochemistry but he taught himself programming/hacking/etc., and with the "right" connections, he was able to enter the field of data science. Now he has been working in the field for quite some time, and managing data science/engineering teams. So if you manage to get your foot in the door and obtain the right experience, your degree may not even matter that much.
But if you're an international student, then it is *much* easier to get an industry job in the U.S. with a PhD. This is because it is easier to get an H1B visa with a doctorate rather than only a Masters.

Great response by @Stat Assistant Professor. 
Casella Berger is already assumed knowledge for some top programs. But if you are admitted based on your pure math background (like yours truly) you likely won't have even cracked open Casella Berger or have taken a proper mathematical statistics course before coming in. However (and I hope this is not too strong of an opinion), Casella Berger presents math stat in a really outdated way. More modern and useful texts nowadays are from Van der Vaart, Lehmann and Casella, Iain Johnstone's Gaussian sequence model book, etc.. IMO Stanford does their math stat sequence in the best and most modern way. Their lectures and homeworks are online. The last part of their sequence, STAT 300C focuses on multiple testing, which is a very hot topic nowadays. 
Also, "hardness" of a program is a really subjective thing. We only discuss about the coursework and preliminary exam requirements above. For me, what also constitutes a big chunk of "hardness" is whether you'll be able to find a strong advisor that you like, whether you'll be allowed to start research asap etc, etc.