(a) Suppose
that the production line is shut down immediately after 60 defective goods have
been produced.
i.
Find the expectation for the total
number of goods produced, proving any formula used (or giving a direct proof).
i.
Find the standard deviation for the
number of non-defective goods produced (you may cite a formula, if you wish).
(a) Suppose
that exactly 1200 goods are produced instead. Using Central Limit Theorem, with
justification, find the probability that at least 70 defective goods are
produced.
Question 2
Consider two continuous
random variables X and Y whose joint density is defined by
(a)
Determine the probabilities P(X ≤ 1 2
, Y ≤ 1 4 ) and P(X ≤ Y ).
Question 3
Suppose X is a discrete
random variable with range {0, 1, 2}, such that
(a)
Determine the value of the constant
d.
(a)
Calculate
for
each value 1 ≤ n ≤ 6.
(a)
Can checking covariance be used to
prove the independence of two random variables?
Now checking the
covariance that can be used to prove the independence of two random variables
and it can be measured by followed by
(a)
Are random variables X,
and
mutually independent? Justify your answer.
No the selected random
variables are not independent as
Question
4
Consider
the following newspaper extract: ”Derby and Nottingham have played 2000200
football matches against each other in total, each game either at Derby or at
Nottingham, and either in sunny or rainy weather. Derby wins more often in
rainy weather than sunny weather, in each city. Derby wins 30% of its games
played in rainy Derby. Nottingham wins or draws 50% of its games played in
sunny Nottingham. A million games took place in rainy Derby and a million games
took place in sunny Nottingham.”
(a) Partition
the sample space of 2000200 ’football match results’ into four appropriately
chosen sets B1, B2, B3, B4, clearly defining each set.
(a) Constructing an annotated 2 by 2 grid, or
otherwise, list the four entries of the form (P(Bi), P(Derby wins|Bi)) for each
value 1 ≤ i ≤ 4. Invent any reasonable values for each piece of information
missing from the newspaper extract, (if any missing information exists).
(a) Calculate
P(Derby wins).
(a) Calculate
P(Derby wins | it is sunny).
(a) Calculate
P(it rains | Derby loses).
(g)
Does rainy weather favour Team Derby? Explain any apparent contradictions
produced by your answers to the previous sections.
Considering
the calculated probabilities for the team Derby it can be concluded that
probability of loss in rainy weather is relatively high for team Derby. On
contrary, probability of win in rain is relatively low for team Derby.
Question
5
Suppose
that whenever you have an odd amount £(2k + 1) of money, you bet £(k + 1) on a
dodgy coin toss (0.52 probability of tails, 0.48 probability of heads). Each
time, you double your bet as profit with a heads, and lose your bet with tails.
Suppose that whenever you have an even amount £(2k) of money, you bet £k on a
similar coin toss. You start with £3. As soon as you have £0 money left, you
stop. Also, if you have more than £6 money, you stop gambling.
(a)
Model the problem as a directed graph
with probability-edgeweights. Make sure that the arcs exiting each state have
weights summing to 1.
Considering the property of the vertices
that are event of bet and edges. The probability of some event becomes equal to
the sum of all the weights and then start the event. The reason for the closure
property is that the model provides conditions for winning money and losing the
toss can cause reduction in winning the money.
(a)
Give the transition matrix for the
Markov chain defined in (a).
(a)
Using (b), give the probability of
having £2 after two coin tosses.
The transition matrix
for the current situation becomes
(a)
List the communication classes of the
Markov chain, and for each class, determine (with proof) whether the
communication class is recurrent or transient.
The
communicability of the Markov chain is based on different state that are state
q and p. the states are equally accessible from the state “i” and state “j”.
the communicability of the states depends on different conditions that are
reflective, symmetric, and transitive. All the conditions are verifiable and
authentic in present state. Considering the transition probability matrix there
are two states that are
.
(a)
Does the Markov chain have a
time-reversible distribution? Prove or disprove.
The Markov chains
are under time reversible distribution if
there is a specified measures
on
S that can satisfy the detailed balanced equations as mentioned below
The present invariant measure of the
chain summarized that
is
infinite and can be normalized under the stationary distribution of the chain.
(b)
Does the Markov chain have a unique
stationary distribution? Prove or disprove.
The random variable shows two values for
winning and loss of coin toss. Consider the geometric distribution that satisfy
the memoryless property under the condition,
the
unique solution for the memoryless property of the exponential random variable
shows that the Markov chains have a unique stationary distribution.
(c)
Calculate the probability of the game
ending with you having more than £6.
Probability
of game ending with 6
