ISS (Statistical Services) Statistics Paper III: Questions 90 - 96 of 96

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Question number: 90

» Applied Statistics » Index Numbers » Fisher Index Numbers: Chain Base Index Number & Tests for Index Number

Appeared in Year: 2014

Essay Question▾

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Define Fisher index number. Show why Fisher index number is said to be ideal index number. Also, show why Laspeyres’ and Paasche’s index number are not ideal one.

Explanation

Fisher’s index number:

Fisher’s index number is the geometric mean of laspeyer’s and Paasche’s index number. This index number uses bith base and current year quantities as weights. It counter balances the effect of upward and downward bias experienced with the method os Laspeyer’s and Paasche’s by taking into account both the current year’s and

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Question number: 91

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

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What is a Wiener process? Obtain the forward diffusion equation of a Wiener process. Also discuss any two application of the process.

Explanation

A stochastic process is a random process that is a function of time. Brownian motion is a stochastic process that evolves in continuous time, with movements that are continuous. So, Brownian motion is a continuous stochastic process, Z (t), with the following characteristics:

  • Z (0) =1

  • Z (t) is continuous.

  • Z (t + s1) −Z (t) is independent

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Question number: 92

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

Short Answer Question▾

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Discuss a one-dimensional random walk.

Question number: 93

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

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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 t

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Question number: 94

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

Essay Question▾

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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)

Since these are independent

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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

Let Zi

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