1.
Page 352 number 1.
From the state
only transitions to
or
are
possible. The rate for the former transitions is the product of the rate at which
mating occurs, namely
male offspring, namely,
rates:
Initial state
times the probability that the mating prod
1.
page 15, number 4.
1.
2.
3.
4.
5.
6.
7.
8.
2.
page 16, number 13.
Let
be the event you roll a on the first toss. Let
be the event that the
first roll after the first toss which is either a 7 or the same as the first toss is the
same as the first toss.
1.
Consider the following strategy for comparing two medical treatments, say
treatment A and treatment B. Patients are treated one at a time and the result of
each treatment is recorded as a Success or a Failure. Every time a treatment
succeeds the next p
1.
A Markov Chain has state space
and transition matrix
Identify all the communicating classes and say whether or not each is transient.
[5 marks]
Solution: Since 2 leads only to 2 one class is
and must be in a class of its own,
. Similarly 3 leads only t
1.
Page 579 number 2.
If
and
are independent and
the conditional distribution of
and variance
is normal with mean
; see the lecture notes.
To apply the result let
that
then
be
is independent of
and
and
be
so that conditioning on
. Note
does
not change the
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Uu'Dtutt%9'`g%S f#%ts
c ne i a s
TV s c a e T cs d V
YfEeW'tfuT
cs d c
tf%S b h 'gUBfg%t'tuutf%)Dgf91'tuutf%)Dgf~UWHD%B9UugfUS P '
x T e b c c q c n e a i a T cV eVsV r S r a q d c X c T cV eVsV r S r a q d c a T e a r S T b aV
1.
Page 234 Number 46.
Solution: To avoid confusion let
labelled
. Let
be the number of sites on the circle,
be the state visited at time
. Let
be the event
whose probability we are to calculate. Consider first the case where
Any trajectory of this kind i
1.
page 85, number 25, page 85, number 26 and page 87, number 40.
25
7 games are played if, after 6 games, each team has won 3. This has probability
This is maximized when
is maximized. Take the derivative
wrt to get
which is 0 when
so this is a maximum.
1.
Cosmic rays arrive at a particle detector according to a Poisson Process with
rate 2.
1.
Compute the probability that 4 cosmic rays arrived in a time period of
length
starting from time
given that given that 8 cosmic
rays arrived in the time period fro
1.
Page 226 numbers 2 and 3.
has eight states
Notice that the first two letters in state
state
must match the last two letters in
because they refer to the same days.
For the states in the order above:
2.
Page 226 number 5.
Notice that
must have
both corr
Lecture 17: More on Hidden Markov Models
(Text Section 4.11)
Recall: Let Yt be the observed response at time t, and let Zt be the hidden state at time t,
t = 1, . . . , n. The process cfw_Yt follows a HMM if
1. There are K hidden states (K is assumed kno