Solutions


Self-check Exercise

A storyteller is given a set of 14 characters. Each night she is to tell a story involving 4 of the 14 characters, using each combination of 4 only once. For how many nights can she continue without repeating any selection of 4 characters?
Hint: "...But when it was midnight Shaherazad awoke and signalled to her sister Dunyazade who sat up and said, "Allah upon thee, O my sister, recite to us some new story, delightsome and delectable wherewith to while away the waking hours of our latter night." "With joy and goodly gree," answered Shaherazad, "if this pious and auspicious King permit me." "Tell on," quoth the King who chanced to be sleepless and restless and therefore was pleased with the prospect of hearing her story."

Exercise

First, set up the plot window to hold 4 plots using par(mfrow=c(2,2)). Then, plot the binomial densities for the following values of n and p: 
     n    p
    20   1/2
    20   1/4
    20   3/4
    20   9/10
Where is the mode located in each case? What is the relationship between the mode and the values of n and p?
The mode is located in the vicinity of n*p.

Exercises

For a random variable X with a binomial(20,1/2) distribution, find the following probabilities.
  1. Pr(X < 8) = 0.1315880
  2. Pr(X > 12) = 0.1315880
  3. Pr(8 <= X <= 12) = 0.736824

For a binomial(200,1/2) distribution:

  1. Pr(X < 80) = 0.001817474
  2. Pr(X > 120) = 0.001817474
  3. Pr(80 <= X <= 120) = 0.996365

For a binomial(2000,1/2) distribution:

  1. Pr(X < 800) = 1.162645e-19
  2. Pr(X > 1200) = 1.162645e-19
  3. Pr(800 <= X <= 1200) "=" 1
    Notes: with about 16 significant digits of accuracy, 1 - 2*1.162645e-19 rounds to 1. If you compute Pr(X > 1200) by subtraction (ie 1-pbinom(1200,2000,1/2)) you will get 0, for the same reason.

September 2001