STAT 135  SPRING '12
A. Adhikari

PROBABILITY PREREQUISITES

Please make sure that you have a solid grasp of the following ideas and results from your probability class.  I have included references from Probability by Jim Pitman and from the early chapters of our own class text (Mathematical Statistics with Data Analysis, by John Rice). Many of you will have used Pitman's book as the text in your Stat 134 class.  I have added references from Probability Theory by Hoel, Port, and Stone (I will call it HPS), as a few of you have used that text.  If you have used yet another text, please look up the index of that book to find the topics below.

Expectation: properties (especially linearity), and the method of indicators. Pitman pages 180-181 are excellent as is the method of indicators on page 168; Rice Section 4.1 is long but it contains all you need.  In HPS see Theorem 2 of Section 4.2.  I don't see the method of indicators in that text.

Variance: properties, especially variances of linear transformations, variances of sums and the relation with independence, and the square root law.  Pitman Section 3.3 is great and has diagrams that indicate why many of the results are true.  Rice Section 4.2 also has everything but is less visual.  HPS has the definition and a short discussion of variance on pages 93-94, variances of sums in Section 4.4 on pages 96-99, and Chebychev's inequality in Section 4.6 on pages 100-101.  See Example 12 on page 98, on the hypergeometric distribution. See how they write the finite population correction.

If you have access to Pitman's text, then to test your understanding please go over the calculation of the mean and variance of the hypergeometric distribution on pages 241-243. This will remind you of the finite population correction.

Also please study the Summary of Chapter 3, pages 248-249 of Pitman.

Covariance:  this appears in the calculation of the variance of a sum of dependent random variables.  Pitman 6.4 (pages 430-431), Rice Section 4.3 (pages 138-139), HPS Section 4.5, pages 99-100.

The Law of Large Numbers (a.k.a. the law of averages):  As n increases, the average of n i.i.d. random variables converges (in a particular sense, and under certain conditions) to the common expectation of the variables.  Pitman page 195, Rice page 178, HPS page 102.  Going over this is especially useful because it forces you to recall the expectation and variance of the mean of an i.i.d. sample.

THE CENTRAL LIMIT THEOREM (yes, it's in bold upper case enlarged font).  The normal approximation to the probability distribution of the sum of a large number of i.i.d. random variables with finite variance. Pitman's treatment is, in my view, more accessible than Rice's.  Read Pitman page 196 and the example on page 197.  Go over the pictures on pages 199-201.  Then read page 268. Go back and thoroughly absorb the special case of the normal approximation to the binomial, on pages 98-102. 
   HPS has a clear treatment in Section 7.5, pages 183-187. 

Change of variable for densities: The density of a function of a random variable whose density you already know. The most commonly used case in Stat 135 is the linear function. Read the para above the box at the bottom of page 302 of Pitman, and then the formula in the box will be easy to rederive from first principles should you forget it. One-to-one non-linear functions are covered on page 304; see page 306-307 for what to do when the function is not one-to-one. The diagrams are very helpful for understanding the derivations, and if you remember the picture clearly then you can rederive the formula yourself if necessary.
   Rice has the general one-to-one case in Proposition B on page 62. HPS does it page 119. The special case of the linear function is in Example 8 on page 121. Neither of these texts provides a diagram.

Special distributions: I expect you to be thoroughly familiar with properties of the following distributions: Bernoulli (that is, the distribution of an indicator), binomial, hypergeometric, geometric, negative binomial, uniform (both discrete and continuous), exponential and gamma, beta, normal (in one and two dimensions).