9.9 Exercises on Chapter 9
Find the value of
by crude Monte Carlo integration using a sample size of n
=10 values from a uniform distribution U(0, 1) taken from tables of random numbers [use, e.g. groups of random digits from Lindley and Scott (1995, Table 27) or Neave (1978, Table 8.1)]. Repeat the experiment ten times and compute the overall mean and the sample standard deviation of the values you obtain. What is the theoretical value of the population standard deviation and how does the value you obtained compare with it?
Suppose that, in a Markov chain with just two states, the probabilities of going from state i
to state j
in one time unit are given by the entries of the matrix
in which i
represents the row and j
the column. Show that the probability of getting from state i
to state j
time units is given by the t
th power of the matrix
Deduce that, irrespective of the state the chain started in, after a long time it will be in the first state with probability and in the second state with probabi-lity .
3. Smith (1969, Section 21.10) quotes an example on genetic ...