What is branching process? Give two real examples of branching process.
If )s(P and P )s( n respectively is the probability generating function (pgf ) of the i.i.d.
random variables and the random variables
Then show that:and
A branching process (Galton–Watson process) is a discrete-time stochastic model for the size of a population in which each individual in generation $n$ produces a random number of offspring independently according to the same offspring distribution, and the members of generation $n+1$ are exactly the offspring of generation $n$. Formally, if $X_n$ denotes the population size at generation $n$ and $\{\xi_{n,i}\}_{i\ge1}$ are i.i.d. offspring _______ ________ ____ _____ _____ ______ ____ __________ _____ _____.
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In a certain state, 58 landfills are classified according to their concentration of three hazardous chemicals: arsenic, barium and mercury. Suppose that the concentration of each one of the three chemicals is characterized as either high or low. If a landfill is chosen at random from among 58 landfills, given the following configuration:
|
| Barium | |||
| High Mercury | Low Mercury | |||
| Arsenic | High | Low | High | Low |
| High | 1 | 3 | 5 | 9 |
| Low | 4 | 8 | 10 | 18 |
It is claimed that the function is the joint 16
distribution function of the random variables X and Y. Then:
i) determine the corresponding joint probability density function fx,y and
ii) calculate the probability
In an investigation related to a specific type of scores of men and women aged 65 to 70, the mean verbal and performance scores for 101 subjects were found to be:
The sample covariance matrix of the scores was
In order to test the null hypothesis that observations came from a population with mean
apply a suitable test statistic. You may consider α = 01.0 for the test.
What is branching process? Give two real examples of branching process.
If )s(P and P )s( n respectively is the probability generating function (pgf ) of the i.i.d.
random variables and the random variables
Then show that:and
For the (M/M/K) : (∞ / FIF )0 queuing model with arrival rate λ and service rate per service channel µ, obtain the steady-state probability of n customers in the system, P . n Hence or otherwise, find the probability that an arrival has to wait.
What is branching process? Give two real examples of branching process.
If P(s) and Pn(s) respectively is the probability generating function (pgf) of the i.i.d. random variables and the random variables {Xn}, where Xn+1 =
Then show that:
Pn(s) = Pn-1(P(s))
and Pn(s) = Pn-1(P(s)).
In a certain state, 58 landfills are classified according to their concentration of three hazardous chemicals: arsenic, barium and mercury. Suppose that the concentration of each one of the three chemicals is characterized as either high or low. If a landfill is chosen at random from among 58 landfills, given the following configuration:
| Barium | ||||
| High Mercury | Low Mercury | |||
| Arsenic | High | Low | High | Low |
| High | 1 | 3 | 5 | 9 |
| Low | 4 | 8 | 10 | 18 |