ISS (Statistical Services) Statistics Paper I (Old Subjective Pattern): Questions 53 - 59 of 165

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Question number: 53

» Probability » Characteristic Function

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Appeared in Year: 2015

Essay Question▾

Describe in Detail

Obtain the characteristic function of X whose pdf is

Explanation

The characteristic function of X is

Let assume x-µ=v, then dx = dv

… (195 more words) …

Question number: 54

» Probability » Expectation

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Appeared in Year: 2013

Essay Question▾

Describe in Detail

Let X be a continuous random variable and have F (x) as the distribution function. If E [X] exists, then show that:

Explanation

We known that in continuous random variable, the mean is

We know that F (x) =1-S (x) and f (x) =dF (x) /dx

dF (x) =-dS (x)

… (77 more words) …

Question number: 55

» Statistical Methods » Non-Parametric Test » Wilcoxon

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Appeared in Year: 2010

Essay Question▾

Describe in Detail

The following data represent lifetimes (hours) of batteries for two different brands:

Brand A

40

30

40

45

55

30

Brand B:

50

50

45

55

60

40

Test whether two brands are the same.

Explanation

we use Wilcoxon two-sample test to the following data

Let n 1 be the number of breakdown times in the Brand A with median time m 1, and n 2 the number of breakdown times in the Brand B with median time m 2. Then the hypothesis is that the median times of the two brands are same against the alternative that they are not same.

… (144 more words) …

Question number: 56

» Probability » Expectation

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Appeared in Year: 2012

Essay Question▾

Describe in Detail

Let X 1, X 2, ···, X n be random variables such that

Also

Find

Explanation

To find the mean and variance, we use the conditional expectation and conditional variance

By the given definition of conditional expectation

… (62 more words) …

Question number: 57

» Probability » Standard Probability Distributions » Normal

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Appeared in Year: 2011

Essay Question▾

Describe in Detail

Let X 1, X 2, …, X n be independent N (0, σ 2) random variables. Obtain the mean and variance of . What is its probability distribution?

Explanation

Let X ~ N (0, σ 2). The density function is

First we find the distribution of Y = X 2

… (118 more words) …

Question number: 58

» Probability » Characteristic Function

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Appeared in Year: 2013

Essay Question▾

Describe in Detail

Find the density function of X whose characteristic function Φ (t) is given by:

Explanation

Rewrite the characteristic function Φ (t)

The density function of X is given by Fourier inversion theorem usin

… (84 more words) …

Question number: 59

» Statistical Methods » Tests of Significance » Chi-Square

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Appeared in Year: 2010

Essay Question▾

Describe in Detail

For 2X2 table

A

b

C

d

prove that Chi-square test of independence gives

Explanation

Let the contingent table is

Class

A

α

Total

B

a

b

a + b

β

c

d

c + d

Total

a + c

b + d

a + b + c + d =N

Here A denotes the presence of any attributes and α denotes the absence of attributes A

Since the marginal frequencies is fixed, therefore

Probability for a indivi

… (88 more words) …