ISS (Statistical Services) Statistics Paper I (Old Subjective Pattern): Questions 55 - 61 of 165

Get 1 year subscription: Access detailed explanations (illustrated with images and videos) to 165 questions. Access all new questions we will add tracking exam-pattern and syllabus changes. View Sample Explanation or View Features.

Rs. 550.00 or

How to register?

Question number: 55

» Statistical Methods » Non-Parametric Test » Wilcoxon

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

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

.

.

.

The variance is

… (182 more words) …

Question number: 57

» Probability » Standard Probability Distributions » Normal

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

Let

The limit is also change

Differentiate with respect to y,

So

… (204 more words) …

Question number: 58

» Probability » Characteristic Function

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 using the characteristic function,

The general form is

Putting the characteristic function

… (159 more words) …

Question number: 59

» Statistical Methods » Tests of Significance » Chi-Square

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 individual belong

… (87 more words) …

Question number: 60

» Probability » Standard Probability Distributions » Uniform

Appeared in Year: 2010

Essay Question▾

Describe in Detail

Let X be a random variable with a continuous distribution function F. Show that F (X) has the uniform distribution on (0,1).

Explanation

Let U = F (x), then the distribution function of G of U is given by

The inverse exists. Since F is non-increasing and its continuous

G (u) =F (F -1 (u) ) {F is a distribution of X}

G (u) =u

Then the p. d. f of U = F (x) is given by

Since F is a distribution function takes value in range [

… (59 more words) …

Question number: 61

» Probability » Standard Probability Distributions » Cauchy

Appeared in Year: 2012

Essay Question▾

Describe in Detail

For the Cauchy distribution given by

where k is a constant to be suitably chosen, derive the expression for the distribution function. Hence obtain a measure of central tendency and a measure of dispersion. What are the points of inflexion of the distribution?

Explanation

We choose k as constant that gives the integral over the range x for the density function is equal to one.

The expression of the distribution function is

Assume

The limit is also change.

U = , L= and put

… (260 more words) …

f Page