(a) By drawing a suitable
directed graph with probability edge-weights, model the described scenario as a
three-state discrete-time Markov chain.
Consider the Markov
Chains on S with different probabilities that are 0.7 and 0.3 for the child of
an amateur video gamer and the pro video gamer with the probability of 0.1 and
the gamer probability of 0.5. the matrix of
.
(b) Write down the
transition matrix P corresponding to the Markov chain that you drew in (a).
Now interpreting the
states for Markov chain for the person who repeatedly plays a game
(c) Based on your answer
to (b), calculate the probability that the great grand-child of a non-gamer is
a pro video gamer. (6 marks)
(d) List the
communication classes of the Markov chain that you drew in (a). For each
communication class, determine with justification whether it is recurrent or
transient. Please state in full any relevant general result s or definitions. (6
marks)
(e) Compute, with
justification, a stationary distribution for the Markov chain that you drew in
(a). Is the stationary distribution unique? In the long run, what proportion of
people in Imagination land are pro video gamers?
Question
2
Suppose
that the joint density function for two continuous random variables X and Y is
defined by
a. Are X and Y independent random variables?
Justify your answer?
Question
3
An unfair coin has
probability 2/5 of landing heads. As an experiment, the coin is thrown
repeatedly, until the first time it lands tails, after which the experiment
ends. Suppose that this full experiment is
conducted n times in total, during which the total number of coin throws is N
(this includes all n experiments).
a)
Name a probability distribution that
is related to the number of throws in a single experiment.
Hence proved that
expected number of throws is 5/3.
a)
Now suppose that the experiment is
modified so that it stops only after 500 coins have landed tails, and before
each throw, the thrown coin is selected randomly between two coins (each with
probability 0.5) : one coin has probability 1 8 of landing heads and the other
coin has probability 4 7 of landing heads. By modifying the arguments, you used
in (b) and (c), or otherwise, find the expected total number of throws for the
full modified experiment to be performed 6 times in a row.
Question 4
Here
is a newspaper passage describing Mad Unicorn Disease: ”Recent reports suggest
that Mad Unicorn Disease is on the rise. Every tenth person in the world now
carries the disease, although 25% of the carriers report no symptoms. The
characteristic symptoms for the most recent viral mutation of the disease are
very distinct, so authorities can now safely assume that non-carriers never
report any false symptoms. A hospital-administered test is reasonably accurate,
but not perfect: it detects the disease in every tenth non-carrier.”
a)
Does
the newspaper passage contain enough information to find the conditional
probability that a person has Mad Unicorn Disease, given that the person tests
positive for the disease? If not, please identify what information is missing
and invent any reasonable-looking quantities for any missing value(s) that you
may need.
Considering the given
situation as we know that every tenth person in the world now carries the
disease, although 25% of the carriers report no symptoms. Therefore it becomes
0.25 for the 1 value.
a)
Based on your answer to (a) (if
needed), calculate the conditional probability that a person has Mad Unicorn
Disease, given that the person tests positive for the disease.
Consider probability
measure for a person who has Mad Unicorn Disease, given that the person tests
positive for the disease, the equation becomes
a)
Based on your answer to (a) (if
needed), calculate the conditional probability that a person has Mad Unicorn
Disease, given that the person reports no symptoms.
Considering
the conditional probability and conditions for the positive and negative
symptoms. The possibility of person with no reports is high as 25% were
reported as persons with symptoms.
a)
Based on your answer to (a) (if
needed), calculate the conditional probability that a person tests positive for
Mad Unicorn Disease, given that the person reports no symptoms.
Considering the
situations of question, a it gives
