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
Describe in Detail
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
Describe in Detail
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 + s_{1}) −Z (t) is independent…
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Question number: 92
» Applied Statistics » Time Series Analysis » Discrete Parameter Stochastic Process
Write in Short
Discuss a onedimensional random walk.
Question number: 93
» Applied Statistics » Time Series Analysis » Discrete Parameter Stochastic Process
Describe in Detail
Define a Poisson process. Stating the regularity conditions, Show that P_{n} (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 nonnegative integer values continuous time process. Assume that

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

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
Describe in Detail
If X_{n} is a branching process with
and σ^{2}= Var (X_{1}), then show that

E (X_{n}) =m^{n}

Explanation
Let X_{0}=1. It is evident
Let Z_{ij} be i. i. d with the offspring distribution P [Z_{ij}=k] =p_{k}, k = 0, 1, 2, . . Such a process {X_{n}} is called branching process and X_{n} denotes the number of individual in the nth generation.
(i)
Since these are independent
By the definition of branching process
In the above equation and the process are identica…
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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
Describe in Detail
Define a Galton Watson branching process. If {p_{k}} is the offspring distribution, then prove the following identities:
(i) P_{n} (s) = P_{n1} [P (s) ]
(ii) ) P_{n} (s) = P [P_{n1} (s) ]
Explanation
Consider a population consisting of individual, able to produce identical offsprings of the same type and dies after producing. Let Z_{ij} be the number of offspring produced by the j^{th} individual of the i^{th} generation and X_{n} be number of individual of the n^{th} generation, j = 1, 2, …, X_{i} and i = 0, 1, 2, …
Let X_{0}=1. It is evident
Let Z_{ij} be indepen…
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