STAT 511 -- EXAM 3 REVIEW SHEET 

I.  Review of some Basic facts about Continuous Distributions (Recall these topics)
 A.  Continuous Random Variables
  1.  Support of a r.v.
     a.  What characterizes the support of a continuous r.v.?
 B.  Continuous Probability Distributions
  1.  Definition of the cdf of a r.v.
  2.  Probability density function (pdf)
     a. Relationship between cdf and pdf of a continuous r.v.
     b. Finding probabilities using the pdf
     c. Solving for a constant that yields a valid pdf
 C.  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 

II. Special Continuous Distributions
 A.  Uniform Distribution
  1.  Using the Uniform pdf to obtain probabilities
  2.  The nature of the Uniform cdf
  3.  Using the Uniform mean and variance
  4.  Relationship between a Poisson Process and the Uniform Distribution
 B.  Moment-generating Functions for Continuous r.v.'s
  1.  Definition of moments of a continuous r.v.
  2.  Definition of moment-generating function of a continuous r.v.
  3.  Process for deriving specific moments using the mgf
 C.  Normal Distribution
   1. Definition and interesting properties of the normal pdf
   2. Using the normal mgf to obtain moments of a normal r.v.
   3. Definition of a Standard Normal r.v.
   4. Standardizing any Normal r.v.
   5. Using Table 4 to obtain normal probabilities
 D.  Gamma and Related Distributions
   1. Support of gamma pdf
   2. Definition and interesting properties of the gamma pdf
     a. Role of shape and scale parameters
     b. Understanding the Gamma Function
   3. Using and recognizing the "kernel" of the gamma pdf
   4. Using the gamma mean and variance
   5. Recognizing a gamma mgf
   6. A special case:  The Chi-square Distribution
     a. Which gamma parameters produce a Chi-square Distribution?
     b. Mean and Variance of a Chi-square Distribution
   7. An important special case:  The Exponential Distribution
     a. Which gamma parameters produce a Exponential Distribution?
     b. Mean and Variance of a Exponential Distribution
     c. Finding exponential probabilities using the exponential cdf (easiest)
        or direct integration (still fairly easy)
     d. Memoryless property of Exponential distribution
   8. Connection between a Poisson process and the exponential (and gamma) distributions
 E.  Beta Distribution
   1.  Support of Beta pdf
   2.  Beta pdf and how its parameter values affect the pdf's shape and skewness
   3.  Using the beta mean and variance
   4.  Finding beta probabilities using the formula involving Binomial probabilities

III. Other Distributional Topics
 A.  Tchebysheff's Theorem
  1.  Markov's Inequality
  2.  Using Tchebysheff's inequality
  3.  Probability statements based simply on a r.v.'s mean and standard deviation
 B.  Piecewise Functions of a r.v. and Mixed Distributions
  1.  Finding Expected Values of a Piecewise Function g(Y)
  2.  Understanding what a Mixed Distribution is
  3.  The cdf of a mixed distribution in terms of a discrete cdf and a continuous cdf
  4.  The "mixing constants" c1 and c2
  5.  Determining the "appropriate" discrete r.v. Y1 and continuous r.v. Y2
  6.  Expected value of a mixed distribution in terms of expected values of a discrete r.v. and a continuous r.v.
  7.  Expected value of a function of a r.v. having a mixed distribution

IV. Bivariate Distributions
 A.  Understanding what a random vector is
  1.  Joint pmf of two jointly discrete r.v.'s
  2.  Joint cdf of two jointly discrete r.v.'s
  3.  Joint pdf of two jointly continuous r.v.'s
 B. Finding Probabilities with Joint pdf's
    1. Integrating the joint pdf over a particular region
    2. Understanding Double Integrals and the correct limits of integration
    3. Sketching the region of support for two jointly continuous r.v.'s
    4. Determining (and sketching) the region of integration to find a certain probability