Two Math Problems

Sometimes I get very frustrated with the master's program I'm in.  Students who actually go to the university, as opposed to my taking classes by watching videos over the internet, have three advantages that I don't:
1)  they can ask questions in class,
2)  they can work with other students in the class to clarify things, or
3)  they can go to the professor's office hours.
I can't do any of those things.  I'm supposed to be able to learn everything just by watching videos of class sessions--and was told this by my adviser.  It's not quite what I thought I was signing up for, but it's what I have.

I have really struggled the last month or so.  I understood the last week's material reasonably well, but everything before that is just a mess in my mind.  I emailed my instructor for some clarification on a specific topic but received no response.  Lacking that big picture understanding, I'm reduced to memorizing some formulas (while not really knowing what they mean or do) and hoping I can figure out how to apply them, and correctly.  I'll be honest, I think the stress of this is partly what caused my shingles outbreak several weeks ago.

Two of the problems from Homework #9 really had me stumped.  So clueless was I that I didn't submit Homework #9.  Instead I kept moving, watched more classes, and completed Homework #10.  Until today I hadn't touched those two problems for about a week.

This morning, though, I decided to try them again; after all, I have to take a test over this material after school on Monday the 9th, and the professor mentioned that one of the test problems would be directly lifted from one of the four homework assignments covering the test material.  Wouldn't it just be my luck that that problem would be one of the two I couldn't do at all?  I had to give it one more go; if I couldn't do them I'd submit the homework assignment incomplete.

One of the problems was an odd-numbered problem so the answer was in the back of the book; I saw no freakin' way to get that answer.  I couldn't even figure out how the variances of the random variables got involved in the answer!  But remember, I've had a week to mull this problem over in my subconscious, and when I looked at it an idea popped into my head.  And when I'd rewritten the problem utilizing that idea, I saw that I could take another step.  But then I stopped, where to go from here?  A few more minutes of scanning the textbook and my class notes didn't reveal anything obvious I'd missed, but then I had another "creative" thought--what if I rewrote it this way?  In effect I added a quantity and then subtracted the same quantity, but this allowed me to do some factoring--and that brought the variance into play!  A few simple algebra steps after that and I had the same answer as in the back of the book!

Where did those "leaps" come from, those ideas, those "let's try this completely new tack" epiphanies?  I don't know.  Maybe we're born with the ability, maybe after some point we've done enough math that they're available to us in the back of our minds.  Maybe it's all just luck.  I don't know, but it took me a couple weeks of trying, and then a week off, before I was able to make those leaps.

So I went to the other problem.  It was an even-numbered problem so I didn't have the actual solution to jump-start any thoughts I hadn't already had.  I stared, I searched for clarification, I did all the things I'd done previously that had failed me.  Then I decided to try some brute force and ignorance.  I went back to a definition, which can sometimes be a very difficult way to solve a problem, but it got me a step further than I'd been able to go beforehand.  It looked like progress!  I kept going, sticking to the definition.  I got an answer of 1, which just doesn't look correct to me, but I'm sure the work I was able to do today got me closer to the answer than I was when I had quit for a week.

I scanned Homework #9 and #10 and emailed them in for grading.  It's a partial victory.