Preparation for Exam 2, Tuesday August 14. 

The exam covers what we've done in 
Devore Chapters 4 and 5 and on the handouts. 

There will be no calculators. You may have to do some 
simple arithmetic.  You will be given a copy of any table 
of values you need for probability distributions. If you 
don't have a table, put numbers into an appropriate formula.

          Continuous random variables

A random variable X is continuous if for every number X, 
the probability that X=x is zero. To understand the relative 
likelihood of values in this situation, we consider the 
likelihood that X takes values in a given interval (a,b). 
In Stat 400 we consider continuous rv's X such that 
there is a function f such that 
   Prob(a<X<b) = integral of f over the interval [a,b]. 
This f is called a probabilty density function (p.d.f.) for X. 

	 Continuous random variables with p.d.f. f

Know when a pdf f is legal 
  (f is nonnegative and total area under the graph of f is 1). 
Know when F is a legal cdf for a continuous r.v.
  (F is continuous, nondecreasing, 
F(x) goes to 0 at -infinity and goes to 1 at infinity).

Know the definition of Prob (a<X<b) in terms of f 
     (the integral of f from a to b) 
and in terms of F 
     (this is just F(b) - F(a)). 

F'(x) = f(x) if F'(x) is defined. 
Be able to compute formulas and graphs for the cdf F given the
pdf f, and vice versa. 

Given the p.d.f. f (or cdf F)  
-be able to compute the expected value E(X) 
  (it is the integral of xf(x)). 
-be able to compute the expected value E(h(X))
 when h is a function of x 
 (it is the integral of h(x)f(x)).

Note: 
E(h(X)) will in general not be h(E(X)). 
For example, consider h(x) = x^2 and X uniformly 
distributed on [-1,1]. Then 
 h(E(X)) = (E(X))^2 = 0^2 = 0,  but 
 E(h(X)) = E(X^2) > 0 . 

       Exponential(lambda) distribution 

Know definition, pmf, cdf for an r.v. X with this distribution. 

Think of X as the time you have to wait for the next event, where 
events occur at a steady rate of lambda events per unit time
independent of previous events.  Note, here: 
- your expected waiting time does not depend on how long you've 
  already waited; I mean by this that 
       Prob (X > s+t | X > s) = Prob (X > t) 
  for any s,t greater than 0. 
- the average waiting time (the expected value of X) is 
  1/lambda (NOT lambda!) (twice as many events per unit time 
  leads to 1/2 the waiting time) 

Easiest to remember: for rv X with this distribution, for any t>0, 
	Prob(X > t) = exp{-lambda t} . 
From this you have the cdf F(t) = 1 - exp{-lambda t}  
and then the pdf f=F'. 
Remark: notice that 
        Prob(X > t(1/lambda)) = exp{-t} 
which doesn't depend on lambda.  You can think of this as saying 
that after rescaling time to multiples of the average waiting time, 
the exp. distributions are all the same. 

       Uniform Distribution on interval [a,b] 

-Know intuition, pdf, cdf, mean. (Easy!) 
-Be able to compute variance and standard deviation.  
 Here is the easy way. 
 If X has uniform distribution on [0,1], it's not hard to 
 check the variance E(X^2) - (E(X))^2  is 1/12, 
 and therefore standard deviation is square root of 1/12.  
 Since Y = a + (b-a)X  has uniform distribution on [a,b], 
 its standard deviation is then (b-a)x(square root of 1/12). 

       Normal Distribution 

Crucial! The bell shaped curve. Know the pdf. 
Be able to use the normal distribution table to 
compute probabilities for any of the normal distributions 
(not just N(0,1) ). Understand all those normal distributions 
are essentially like N(0,1), recentered at the mean and rescaled 
by the standard deviation. 

Be able to find a percentile of the N(0,1) distribution. 
For example, the 90th percentile is (to a decimal approximation)
equal to 1.645, because if an rv Z has the  N(0,1) distribution, 
the Prob(Z < 1.645) = .90 .  
Be able to find a percentile of an N(mu,sigma) distribution. 
For example, its 90th percentile would be the number 
mu + (1.645)sigma . 
Again, this looks just like N(0,1) -- recentered at mu, 
and rescaled by sigma. 

Know approximation of Binomial(n,p) by normal distribution 
for n large and p, (1-p) not too small: 
   rule of thumb: this apporoximation is ok for 
   both np and n(1-p) at least 10. 
Be able to use the continuity correction for your normal approximation. 

       aX + b 

For a random variable X and numbers a and b, 
know what E(aX+b), Var(aX+b) and st.dev.(aX+b) are in terms of the 
corresponding items for X. If Y is another, independent random variable, 
be able to compute E,Var and st. dev of aX + bY + c.  
Know for independent rvs X,Y that E(XY)=E(X)E(Y). 
Also, if X and Y are independent, then so are are the rvs 
f(X) and g(Y), if f and g are functions from the real numbers 
to the real numbers. 

Understand the website handouts  on CLT and Normal Distributions, 
and in a rough way the LLN. 

Everything in 5.1, and also the Proposition in  5.2 on 
computing expected value, is fundamental. 
Devore has done a good job of putting boxes around the 
summary items you should understand. In particular: 
you should be able to compute an expected value 
of a function h(x,y) of two random variables, and  
you should be able to compute the probability of 
an event defined in terms of two random variables.  
Especially note that the joint density function of 
INDEPENDENT random variables is the product of their 
individual density functions. Be prepared to compute  
double integrals. 

For uniform distribution on a region, probability is 
proportional to area, and this sometimes simplifies probability 
computations. 

For ANY random variables:  
the expectation of the sum is the sum of the expectations.

For INDEPENDENT random variables, we also have:  
the variance of the sum is the sum of the variances. 
For example, if X,Y are independent, then
      E(XY) = E(X)E(Y)      and 
   V(X + Y) = V(X) + V(Y) . 
Note these statements are usually not true without the independence 
assumption. Consider for example the case Y = -X. 
Also note you can use the equation for V(X+Y) to find the standard 
deviation of X+Y (take square root of V).  

The idea of a sampling distribution of a statistic, described 
in 5.3, is fundamental to understanding statistics (though 
somewhat difficult to test on an exam). A statistic is some quantity 
calculated from sample data. The key idea is that the 
statistic can be considered as a random variable itself, 
with its own probability distribution. 

The most important statistic to understand is the average, 
Xbar_n (this is (1/n)(X_1 + ... X_n), where _ denotes subscript). 
We generally consider the case that the X_i are i.i.d. 
(so they can be thought of as representing a random sample from 
a population with an underlying probability distribution).  
Know all about the average as discussed on the CLT and LLN 
handouts. Also: know the statements of CLT and LLN: not only 
verbatim, but also the meaning of the assumptions and conclusions.  

Regarding correlation and covariance: it is sufficient 
to know the boxed items in 5.2 

A linear combination of independent normal random variables is 
still normal. Understand how to compute the mean, standard deviation and 
variance of a linear combination of independent random variables. 
If X is normally distributed, understand how to compute probabilities 
involving X from the table for the standard normal distribution. 
   
Homework and quizzes 

The test will reward understanding of homework and quizzes.