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 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.
Measure theory being necessary to statistics is rather contentious; a better discussion is on Andrew Gelman's blog [1].
My school's PhD stats program does require real analysis before the prelims, but for most intents and purposes, 'multi' and 'linal' (as the cool kids say) should be sufficient for machine learning from a comp sci perspective.
I haven't fully worked through ESLR (Hastie and Tibsharini's advanced version of ISLR posted above) but the majority of the math there is linear algebra with some differential equations and calculus thrown in. I've heard Harvard Stat 210 and Berkeley Stat 205A/B cited as good examples of mathematical stat classes - if you're seriously interested maybe take a look at those syllabi.
The dependency would look like:
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.