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 counts with common distribution $\xi$, then $$ X_{n+1}=\sum_{i=1}^{X_n}\xi_{n,i}, $$ with the convention $X_0$ given (often $X_0=1$). The process models extinction, explosion, or stable behaviour depending on the mean offspring number. Two real examples: _________ __________ ________ _____ _______ ______ ________ _____ ________ _______.
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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 vector apply a suitable test statistic. You may consider α = 0.01 for the test.
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 order to fit the regression line y = β1 + β2 on a data set consisting of 34 pairs of values (z,y), the least square estimates of β1 and β2 are to be computed. From the data set, the following values are obtained:
Obtain the fitted regression line.
Let {Xn; }0 n ≥ be a Markov chain with four states; 1, 2, 3, 4 and the following transition probability matrix:
| 1 | 2 | 3 | 4 | |
| 1 | 0 | 0 | 1 | 0 |
| 2 | 1 | 0 | 0 | 0 |
| 3 | 1/2 | 1/2 | 0 | 0 |
| 4 | 1/3 | 1/3 | 1/3 | 0 |
State whether the following statements are true or false. Justify your answer with a short proof or a counter example:
i) Although in one-step transition probability matrix, P, of a Markov chain the sum of each row must necessarily be unity but in higher order transition probability matrices, P(j); j = 2,3,4,...... This rule is not necessary.
ii) For the joint pdf of random variables (X,Y) given by:
iii) Let the r.v. X follows the negative exponential distribution with parameter . Then
iv) One of the examples of non-Poisson queuing system is the (M/G/1): (∞(F1F0) queuing model.
v) If X1,X2,..., X, be a random sample from Np, (μ, Σ), then maximum likelihood estimators of µ and ∑are:
The joint distribution of the random variables X and Y is given by:
| x /y | −1 | 0 | 1 |
| −1 | α | β | α |
| 0 | β | 0 | β |
| 1 | α | β | α |
| 1 | 2 | 3 | 4 | |
| 1 | 0 | 0 | 1 | 0 |
| 2 | 1 | 0 | 0 | 0 |
| 3 | 1/2 | 1/2 | 0 | 0 |
| 4 | 1/3 | 1/3 | 1/3 | 0 |
i) Find the probabilit
b) It is claimed that the functionis the joint distribution function of the random variables X and Y. Then:
ii) calculate the probability