Course: Probability: Distribution Models & Continuous Random Variables
Length: 6 weeks, 4-6 hrs/wk (ha!)
Instructor: Mark D. Ward
In this statistics and data analysis course, you will learn about continuous random variables and some of the most frequently used probability distribution models including, exponential distribution, Gamma distribution, Beta distribution, and most importantly, normal distribution.
You will learn how these distributions can be connected with the Normal distribution by Central limit theorem (CLT). We will discuss Markov and Chebyshev inequalities, order statistics, moment generating functions and transformation of random variables.
The same comments I made about Part 1 of this course – the part covering basic probability concepts and discrete models – hold for this one: it’s a great course in so many ways, but it’s missing some kind of connective tissue. And the support – that is, forum assistance – is sketchy at best.
I have to smile when I see the expected workload for six weeks is 4 to 6 hours a week. Yet, I can see how that might be true for those who can listen to something like this…
Okay, the first one I’ll tell you about is the Weak Law of Large Numbers: it says that what we should do is fix an epsilon – it’s positive, it’s usually small, maybe you use epsilon as 1/1000 or 1/10000. And consider an infinite sequence of random variables, say X1, X2, X3, etc., that are independent. So then the probability that the average of the first n random variables is more than epsilon away from the mean of the random variables converges to 0 as n goes to infinity.
… and grasp it without parsing over it and remembering the (ε, δ)definition of limits and how it isn’t really that complicated, it’s just really nasty to put into words. Or mathematical notation, for that matter. If you can read Math, you’ll do fine. For the rest of us, it’s gonna take a lot longer, and involve a lot more sweat. But it can be done: In eight 12- to 14-hour weeks, I managed to come out of this with a decent grade, though I have to say, I strongly suspect the deck was stacked to puff up grades. I’m not fooling myself: I have a long way to go before I “understand” this stuff.
Included was what I’m discovering is the standard probability curriculum: various continuous probability distributions complete with PDFs, CDFs, expected values, variance, sums, and conditionals, Markov and Chebychev inequalities, covariance, moment generating functions, and transformations. Again, as with Part 1, each set of lectures is followed by three or four sets of ungraded practice questions in PDF form, and that’s where the real learning takes place. Weekly graded quizzes follow; these are well-designed with both basic-concept questions (“find the expected value of this PDF”) and more complicated problems. There are also several “gimmes” along the way – seriously, “Your answer should be 3”, why is this even a question? And “your answer to d should be the same as your answer to b” gives you two chances to come up with the goods – hence my impression that there’s some padding going on.
The prerequisites recommend three semesters of Calculus (“including double integers”, which presumably should be “double integrals”) and sure enough, many of the problems require integration, a few need differentiation, and infinite series pop up every once in a while. While I can differentiate reasonably well, integration has always been a problem. I found this course helped my precarious understanding of integrals a lot, particularly with things like integrating xy with respect to y, exponentials, u-substitution, and integration by parts (the whole calculate-this thing). For the bulk of the work, I relied on Wolfram Alpha and Symbolab, because I’m picking my battles. So sue me. For my purposes, it worked ok, and even was helpful. I wouldn’t recommend trying this without some prior exposure to calculus, however.
Another way I used this course, besides the obvious learning about probability, was to improve my ability to “read’ math. I’m by nature a reader, but when it comes to math, I look at page of notation which presumably contain their own explanations, and my eyes just glaze over. It’s why moocs are so much more useful to me than textbooks. Here, where each lecture includes a PDF handout which is often the entire lecture, I went through the handouts first, literally copying them into my notes document, and tried to understand what was going on. Then I’d listen to the lectures, which meant two passes over the material. I still struggle with reading math, but it’s a start.
The lectures mentioned a few times “If you have any questions, come and see me” which means this was intended for a flipped classroom, not solo study. That can work really well, but the support wasn’t really there; the discussion forums were empty. I asked three questions, got two answers five days later, and they assumed I was asking different questions. So it was just as lonely as Part 1, which I took in archived form. I’m still shocked that the forums are (after I deleted my posts) empty; isn’t anyone taking this course, or does everyone but me just understand this stuff?
In spite of all my complaints, I still thought this was a great two-part course, just what I needed to provide enough background so I could go back to Harvard’s Introduction to Probability course that I had to put on hold [addendum:yeah, after taking another look, I’ve put this on permanent hold, aka dropped it, not gonna happen, just way way too mathy] when it became evident that I wasn’t getting it (and, by the way, with the exception of a few lead-in videos covering a broad overview of topics, is entirely in written form – hence my need to improve my ability to read math). That’s the benefit of moocs: you can keep taking stuff over and over without fighting with the Enrollment Office or with the bursar. Taking stuff from different profs also offers the benefit of realizing that one person’s “find the density of X” is another’s “find the PDF of X”. It’s an approach I find helpful: the first time through, I get some idea of the lay of the land, and by the second (or maybe the third, or fourth, whatever it takes) time, I’m ready to actually start learning.
There’s a whole other course coming up in May. Maybe then I’ll be able to say I get it. Probably not. But maybe.