you are familiar with the term average, as in arithmetical average of a set of numbers test scores f

1

• you are familiar with the term average, as in
  arithmetical average of a set of numbers (test
  scores for example) we used the symbol to
  stand for this average. In probability, we have
  an average value too the so-called
  mathematical expectation, which is defined
  intuitively as the long-run average value of
  the possible values of an experiment. In
  particular, if the possible values of a
  probability experiment are a1, ... , ak and the
  corresponding probabilities of these values
  occurring are
• p1, ... , pk then the expectation (E) is the
  weighted average of the values, with the weights
  being the probabilities
• Illustrate this with the relative frequency
  approach to probabilities...
• Example Pick 3 lottery payoff is 500, cost
  of ticket is 1
• HW page 95, 3.81-3.86
• Review exercises (p.97-99) 3.98, 3.100-3.104,
  3.107

2

• We are going to be concerned with random
  variables, a function that assigns a numerical
  value to each possible outcome of a random
  experiment
• Xsum of spots when a pair of fair dice is thrown
• Y of Hs that come up when a fair coin is tossed
  4 times.
• We are going to be interested in the probability
  distribution of the random variable (rv) i.e.,
  the list of all the possible values of the rv and
  the corresponding probabilities that the variable
  takes on those values... get the probability
  distributions of the two example rvs X and Y
  above...
• If we write f(x) P(X x) then it must be the
  case that

3

• We will be considering several discrete random
  variables and their distributions
• Binomial X the number of Ss in n Bernoulli
  trials
• (recall Bernoulli trials or see p. 105)... our
  previous rv counting the number of Hs in 4 tosses
  of a fair coin is a typical Binomial variable
  we denote the binomial probabilities by
• Cumulative binomial probabilities are given in
  Table 1 for various values of n and p write
• TI-83 can do binomial probabilities under 2nd
  VARS binompdf (individual binomial probs) or
  binomcdf (cumulative binomial probs)
• R can calculate binomial probabilities (see the R
  handout)

4

• we can think about the binomial rv as sampling
  with replacement (Hs and Ts in equal numbers in
  an urn, replace after each draw...). The
  hypergeometric rv can be thought about in the
  same way except that the sampling is to be done
  without replacement from an urn with N items, a
  Ss and N-a Fs.
• Lot of 20 items, 5 defective (success). Choose
  10 at random what is the probability that
  exactly 2 of the 10 are defective?
• Ans h(2 10, 5, 20) .348
• NOTE This can be approximated by b(2 10, .25)
  .282 (not so good...but if the N were larger...
  what would happen?)

5

• Example (p.111) N100, a25 (so p remains .25)
• then h(2 10, 25, 100) .292 and b(2 10, .25)
  .282
• The mathematical result is that as N approaches
  infinity with pa/N, h(x n, a, N) approaches
  b(x n, p) and a good rule of thumb is to use the
  binomial if n lt N/10
• HW Read sections 4.1-4.3 work on problems
  4.2, 4.3, 4.5, 4.7, 4.9, 4.13, 4.17, 4.19, 4.20,
  4.21, 4.23, 4.25, 4.28, 4.29