# ISS (Statistical Services) Statistics Paper III: Questions 92 - 101 of 109

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## Question number: 92

» Applied Statistics » Time Series Analysis » Discrete Parameter Stochastic Process

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### Write in Short

Discuss a one-dimensional random walk.

## Question number: 93

» Applied Statistics » Time Series Analysis » Discrete Parameter Stochastic Process

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### Describe in Detail

Define a Poisson process. Stating the regularity conditions, Show that Pn (t) =P {N (t) =n} is given by the Poisson law

### Explanation

Let X (t) denote the number of occurrences of a typical event over [0, t], X (t) is also referred as a counting process. Let X (t) be non-negative integer values continuous time process. Assume that

1. X (t + h) -X (t) is independent of X (t) -X (0) with X (0) =0 that process with independent increments.

2. X (t + h) -X (t) does not depend on t that is X (

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## Question number: 94

» Applied Statistics » Time Series Analysis » Discrete Parameter Stochastic Process

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If Xn is a branching process with

and σ2= Var (X1), then show that

1. E (Xn) =mn

### Explanation

Let X0=1. It is evident

Let Zij be i. i. d with the offspring distribution P [Zij=k] =pk, k = 0,1, 2, . . Such a process {Xn} is called branching process and Xn denotes the number of individual in the nth generation.

(i)

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## Question number: 95

» Applied Statistics » Time Series Analysis » Discrete Parameter Stochastic Process

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What is Gambler’s ruin problem? Obtain the probability of ruin, if Gambler plays with a capital of a rupees.

## Question number: 96

» Applied Statistics » Time Series Analysis » Discrete Parameter Stochastic Process

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Define a Galton- Watson branching process. If {pk} is the offspring distribution, then prove the following identities:

(i) Pn (s) = Pn-1 [P (s) ]

(ii) ) Pn (s) = P [Pn-1 (s) ]

### Explanation

Consider a population consisting of individual, able to produce identical offsprings of the same type and dies after producing. Let Zij be the number of offspring produced by the jth individual of the ith generation and Xn be number of individual of the nth generation, j = 1,2, …, Xi and i = 0,1, 2, …

Let X0=1. It is evident

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## Question number: 97

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State and explain various types of data is used in the estimation of the model.

### Explanation

In the estimation of model the following types of data are used:

(I) Time series data:

Time series data give information about the numerical values of variables from period to period and are collected over time.

For example, the data during the years 1990 - 2010 for monthly income constitutes a time series data.

(II) Cross section data:

The cross sectio

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## Question number: 98

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Explain meaning of econometric models.

### Explanation

A model is a simplified representation of a real world process. It should be representative in the sense that it should contain the salient features of the phenomena under study.

In general, one of the objectives in modeling is to have a simple model to explain a complex phenomenon. Such an objective may sometimes lead to oversimplified model and so

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## Question number: 99

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What is National Income?

## Question number: 100

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What are the aims of econometrics? How econometrics differ from economic statistics?

### Explanation

The three main aims of econometrics are as follows:

(I) Formulation and specification of econometric models:

The economic models are formulated in an empirically testable form. Several econometric models can be derived from an economic model. Such models differ due to different choice of functional form, specification of stochastic structure of the v

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## Question number: 101

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State and explain how to estimate Lorenz Curve.

### Explanation

Let X be the outcome variable of interest (e. g. income). The cumulative distribution function of X is given as and the quantile function (the inverse of the distribution function) is given as with For continuous X, the ordinates of the relative Loren

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