Stat 414 Final Exam Study Guide

The comprehensive final exam is scheduled for Wednesday, Dec 17, 2003 from 6:50-8:40 pm in 258 Willard.

You should know:

  1. How to list the elements of the outcome (sample) space (and of possible events) of a random experiment.
  2. The difference between mutually exclusive events, independent events, exhaustive events.
  3. Three different ways of assigning probability to an event (personal opinion, relative frequency, classical)
  4. How to apply the three axioms of probability and the five theorems that followed.
  5. Counting techniques -- multiplication rule, permutations (ordered) of n distinct objects, permutation of n distinct objects taken r at a time, combinations (unordered), permutations of indistinguishable objects.
  6. Definition of conditional probability and the multiplication rule.
  7. Definitions of independent events.
  8. How to apply Bayes' Rule.
  9. How to check to see if a function f(x) is a valid probability mass function. Know specifically for binomial, geometric, and poisson random variables.
  10. How to calculate the expectation of any function of a discrete random variable.
  11. How to calculate the mean, variance and standard deviation of any discrete random variable.
  12. The rules for expectation, mean, and variance.
  13. How to check four conditions for binomial random variable.
  14. How to calculate probabilities, means, and variances for each type of discrete random variable we've studied.
  15. How to find a moment-generating function, especially for the discrete random variables we studied.
  16. How to use a moment-generating function to find a mean and variance or to identify a p.m.f.
  17. The characteristics of a valid p.d.f for a continuous random variable.
  18. That to find the probability that a general, continuous random variable takes on values in some interval, you need to find the area under the curve.
  19. That the probability that a continuous random variable takes on a specific value is 0.
  20. How to find the cumulative distribution function of a continuous random variable.
  21. That the cumulative distribution function of a continuous random variable is a continuous, nondecreasing function which takes on values between 0 and 1, inclusive.
  22. How to find a percentile, quartile, median using either a p.d.f or a c.d.f.
  23. How to determine the expected value, variance, standard deviation, moment-generating function for a general, continuous random variable.
  24. How to use a moment-generating function M(t), or a cumulant-generating function R(t) = lnM(t), to find the mean and variance of a continuous random variable.
  25. The characteristics (derivation, p.d.f., mean, variance, moment generating function) of a uniform, exponential, gamma, chi-square and normal random variables.
  26. Know how to find probabilities for the named continuous distributions we studied.
  27. The interpretation of Z.
  28. The empirical rule.
  29. How to find the distribution of a function of a random variable using either the distribution function technique or the change of variable technique.
  30. How an observation of X with distribution function F(x) can be simulated.
  31. How to create a general q-q plot.
  32. How to verify that a joint p.m.f. (p.d.f.) is valid.
  33. How to check that X and Y are independent.
  34. How to calculate the mean and variance of a random variable given the joint p.m.f. (p.d.f).
  35. That (and why) if X and Y have a triangular support, then they are necessarily dependent.
  36. The trinomial distribution.
  37. How to find probabilities using a joint p.m.f. (p.d.f.).
  38. How to find marginal distributions given the joint p.m.f. (p.d.f.)
  39. How to calculate the covariance and correlation between X and Y.
  40. How to interpret the correlation coefficient.
  41. That Corr(X,Y)=0 does not imply X and Y are independent.
  42. The distinction between a joint p.m.f. (p.d.f) and a conditional p.m.f. (p.d.f.).
  43. How to find a conditional p.m.f. (p.d.f.) and how to verify that one is valid.
  44. How to find a conditional mean and conditional variance.
  45. How to solve every assigned problem.

Know how to prove:

  1. P(A or B) = P(A) + P(B) - P(A and B)
  2. The shortcut formula for the variance (from the definition of the variance)
  3. That geometric r.v. X has "loss of memory." (Section 3.4, problem #12)
  4. Pascal's equation (#2.2.12.)

NOTE: You will also be given the formula sheets in the inside front cover of the textbook, as well as any statistical tables that you might need.