.
Suppose X = X0,X1, . . . , Xn are observations on a Markov
chain with transition matrix P = (pij ) and let n(i, j) be the number of times
that state i is followed by state j in the sequence X. Find the maximum
likelihood estimator of the elements pij in terms of the n(i, j). [Hint: See the
discussion in Section 2.6.3.]
.
Use the value of p computed for HpaII. Compute
λ = 1/p for the parameter of the corresponding exponential distribution, using
the approach of Section 3.3.
a. For the bins [0, 100), [100, 200), [200, 300), [300, 400), [400, 500), [500, 600),
[600,∞), compute the theoretical probability of each bin.
b. Use the probabilities from (a) and the expected number of fragments from
an HpaII digestion of bacteriophage lambda to calculate the expected
number of fragments in each of the seven bins.
c. Compute the X2 value analogous to (2.29) on page 64 for these observedexpected
data. The number of degrees of freedom for the approximate χ2
distribution of X2 is equal to 7 −1 = 6.
d. Does the exponential distribution fit these data?