[mturmon]

I agree with many of the responses here, that Math. Analysis (epsilon-delta proofs, continuity, etc.) is not strictly necessary for statistics. But...it certainly will help.

The dependency would look like:

  stats <- measure-theoretic prob. <- math. analysis

The problem with dumping the measure-theoretic probability is that you won't really know what a random variable is. It has a definition (a measurable function into the reals), and without that, you will have a tendency to think of it as "a box that produces something random when you look into it". This will limit your ability to understand papers, and will make you insecure in talking to people.

Besides "random variable", other common notions will also be hard to understand without measure-theoretic probability, like "almost surely", convergence concepts, the difference between the SLLN and WLLN, etc.

The problem with dumping analysis is that you will not know some basic things like what a continuous function is. What is everywhere continuous? What is a C1 function? And again, you will have a hard time reading and speaking.

For what it's worth, I found analysis to be not that fun, but measure-theoretic probability to be really a fun, tight, theory. It was enjoyable to learn.