STAT 511 -- EXAM 2 REVIEW SHEET 

I. Basics of Discrete Random Variables
 A.  Definition of a r.v.
   1. Discrete r.v.'s
   2. Discrete probability distributions
     a. Representations:  Table, Graph, Formula
   3. Properties of a Valid Discrete probability distribution
 B.  Expected Values
   1. Definition of expected value
     a. Interpretation as a population mean
   2. Expected Value of a function of a r.v.
     a. Expected values involving constants
     b. Expected Value of a sum of functions of a r.v.
   3. Definition of variance
     a.  Definition of standard deviation
     b.  Alternative formulation of the variance
   4. Finding expected value and variance for a given discrete probability distribution

II. Special Discrete Probability Distributions
 A.  The Binomial Distribution
  1.  Definition of a Binomial Experiment
  2.  How does a binomial r.v. relate to a binomial experiment?
  3.  Binomial probability function
     a. Using the binomial expansion formula
  4.  Finding binomial probabilities using the probability formula
  5.  Finding (cumulative) binomial probabilities using Table 1
  6.  Mean of a Binomial r.v.
  7.  Variance and standard deviation of a Binomial r.v.
  8.  The Bernoulli distribution
 B.  The Geometric Distribution
  1.  Sampling structure for a geometric-type experiment
  2.  Geometric probability function
     a. Finding geometric probabilities using the probability formula
  3.  Using formulas for sums of geometric series
  4.  Mean of a geometric r.v.
  5.  Variance and standard deviation of a geometric r.v.
 C.  The Negative Binomial Distribution
  1.  Main difference between NB r.v. and geometric r.v.
  2.  Negative Binomial probability function
     a. Finding Negative Binomial probabilities using the probability formula
  3.  Mean of a Negative Binomial r.v.
  4.  Variance and standard deviation of a Negative Binomial r.v.
  5.  Main difference between binomial and negative binomial r.v.'s
 D.  The Hypergeometric Distribution
  1.  Sampling structure for a hypergeometric-type experiment
  2.  Difference between binomial experiment and hypergeometric-type experiment
  3.  Hypergeometric probability function
     a. Finding hypergeometric probabilities using the probability formula
  4.  Mean of a hypergeometric r.v.
  5.  Variance and standard deviation of a hypergeometric r.v.
  6.  Relationship between binomial and hypergeometric r.v.'s
     a. Using binomial probabilities to approximate hypergeometric probabilities
     b. When is this approximation valid?
 E.  The Poisson Distribution
  1.  Motivation for Poisson as a limit of binomial probabilities 
  2.  Poisson probability function
  3.  Finding Poisson probabilities using the probability formula
  4.  Finding (cumulative) Poisson probabilities using Table 3
  5.  Mean of a Poisson r.v.
  6.  Variance and standard deviation of a Poisson r.v.
  7.  Relationship between binomial and Poisson r.v.'s
     a. Using Poisson probabilities to approximate binomial probabilities
     b. When is this approximation valid?

III.  More about Probability Distributions
 A.  Moments and Moment-generating Functions
  1.  Basic Definition of the k-th Moment
     a.  Some special moments of interest
     b.  Moments about the mean
  2.  Definition of the mgf
     a.  When does the mgf exist?
  3.  Relationship between the mgf and the moments of a r.v.
     a.  How do we use the mgf to get the k-th moment?
     b.  Why does this work?
  4.  Examples of finding mean and variance of various r.v.'s using mgf's
  5. The mgf uniquely characterizing a distribution
     a.  Using the form of a mgf to identify the specific distribution of a r.v.
 B.  Continuous Random Variables
  1.  Support of a r.v.
     a.  What characterizes the support of a continuous r.v.?
 C.  Continuous Probability Distributions
  1.  Definition of the cdf of a r.v.
     a. Finding the cdf of a discrete r.v.
     b. Properties of any cdf
     c. Difference between the cdf of a discrete r.v. and cdf of a continuous r.v.
     d. Finding probabilities using the cdf
  2.  Probability density function (pdf)
     a. Relationship between cdf and pdf of a continuous r.v.
     b. Properties of any pdf
     c. Finding a pdf, if given a cdf
     d. Finding a cdf, if given a pdf
     e. Graphing a cdf and/or a pdf
     f. Finding probabilities using the pdf
     g. Solving for a constant that yields a valid pdf
  3.  Quantiles of a r.v.
     a. Definition of a quantile
     b. Finding a quantile for a continuous r.v.
     c. Median of a continuous r.v.
 D.  Expected Value of a continuous r.v.
   1. Definition of expected value of a continuous r.v.
   2. Expected Value of a function of a r.v.
   3. Definition of variance
   4. Finding expected value and variance for a given continuous probability distribution