Find the differential equation of pure birth process with and the process start with one individual at time t = .0 Hence, find
[N(t) is the number present at time t] with
and
. Also identify the distribution
A pure birth process is a stochastic process that models the population growth where individuals reproduce independently, and the birth rate depends only on the current population size. Let's denote the birth rate as \( \lambda_K = K \lambda \), where \( \lambda \) __________ _________ ________ __________ ____ _____ _______ __________ _________.
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The Tooth Care Hospital provides free dental service to the patients on every Saturday morning. There are 3 dentists on duty, who are equally qualified and experienced. It takes on an average 20 minutes for a patient to get treatment and the actual time taken is known to vary approximately exponentially around this average. The patients arrive according to the Poisson distribution with an average of 6 per hour. The officer of the hospital wants to investigate the following:
i) The expected number of patients in the queue.
ii) The average time that a patient spends at the clinic.
iii) The average percentage idle time for each of the dentists.
The joint density function of random variables X, Y and Z is given as ;
where and
Find
i) the constant K.
ii) the marginal distributions of X, Y and Z
iii) E(X), E(Y) and E(Z)
iv) the conditional expectation of Y given X and Z.
v) the correlation coefficient between X and Y
For the two-state Markov chain, whose transition probability matrix is
Find all stationary distributions
Consider the Markov chain having the following transition probability matrix.
i) Draw the diagram of a Markov chain.
ii) Classify the states of a Markov chain, i.e., persistent, transient, non-null and a periodic state. Also check the irreducibility of Markov chain.
iii) Find the closed sets.
iv) Find the probability of absorption to the closed classes. Also find the mean time up to absorption from transient state 3 to 4.
Find the differential equation of pure birth process with and the process start with one individual at time t = .0 Hence, find
[N(t) is the number present at time t] with
and
. Also identify the distribution
Let pK ,where K = 0,1,2 be the probability that an individual generates K offsprings. Then find the p.g.f. of {pK } . Also, calculate the probability of extinction when
i). and
ii). and
Suppose that the probability of a dry day (State 0) following a rainy day (State 1) is 1/3 and the probability of a rainy day following a dry day is 1/2. Write the transition probability matrix of the above Markov chain.
Given that 1st May is a dry day, then calculate
i) the probability that 3rd May is also a dry day.
ii) the stationary probabilities.