# Any good practice material for expected value and variance?

I am trying to learn more about probability mass functions, density functions, expected value, and variance. Are there any online materials or quizzes (with answers and explanations) that I can use to test my understanding?

## 1 Answer

In my experience teaching intro probability courses, many online instructional materials (even trendy and popular ones) are quirky as to content or hastily written without proofreading. (I have spent many hours over the years dealing with student confusion due to reading poorly written online material.)

I don't want to deprecate online probability instructional materials en masse, but I don't have any suggestions that I can recommend with confidence. If someone else has a recommendation based on personal experience, perhaps it will appear in another Answer.

If you are looking for something that's free online, then I can suggest Grinstead and Snell's 'Intro. to Probability'. It is available in book form, but you can find a free PDF online. Look at Chapter 6.

There are many exercises at the end of the chapter, but I don't know if an answer book is available. However, there are many worked examples. You can choose to view some of them as problems, if you don't peek at the solutions.

Grinstead & Snell is competently written and well-reviewed. It has some examples that are not standard for elementary texts (which was sort of the point of their writing project). Also, the book uses simulation in some places where a traditional text would not. However, there are many very traditional examples, arranged in a reasonable progression of difficulty. I suggest you just skip the examples that are not to your taste. Also, remember that it is Chapter 6, that deals with expectations and variances, so you might want to browse Chapters 1-5 to make sure you have the appropriate background material in mind.

• If you can recommend a good book that I can order from Amazon or something I can go that route as well. I just figured I may as well try the free route first. But if there is a fantastic resource out there I can buy as well, I am all ears
– AJJ
Aug 10 '16 at 19:34
• If you can be explicit about your mathematical level and your reasons for studying probability, you might get a variety of useful recommendations-- before you spend money. There are dozens of possibilities across math level and degree of application. Meanwhile, I think Grinstead and Snell is worth a look. Google Grunstead Snall Dartmouth Probability. Aug 10 '16 at 19:39
• I'm self-taught so I am not sure how to express my math ability. I just want to learn everything I can. To better problem-solve, analyze distributions, analyze data, understand statistical analysis, understand probability better, everything. Number theory, computer science, calculus, probability, everything.
– AJJ
Aug 10 '16 at 19:55
• Differential and integral calculus? Multivariable calculus? Real variables? Measure theory? // Interested in pure math? Grad study in probability? Job in quality management, applied probability, actuarial science, finance? 'Everything' is probably not attainable. Aug 10 '16 at 20:05