ISS (Statistical Services) Statistics Paper I (Old Subjective Pattern): Questions 59 - 64 of 165

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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

Equation

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

… (112 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

Equation

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

… (37 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

Equation

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.

Equation

Equation

Equation

Equation

The expression of the distribution function is

Equation

Equation

Equation

Assume Equation

The limit is also change.

U = Equation , L= Equation and put

… (148 more words) …

Question number: 62

» Probability » Sample Space

Appeared in Year: 2011

Essay Question▾

Describe in Detail

A fair die is thrown until a 6 appears. Specify the sample space. What is the probability that it must be thrown at least 3 times?

Explanation

A fair die content the value {1, 2, 3, 4, 5, 6}

So, the probability of getting 6 is p = 1/6, then probability of getting other than 6 is q = 5/6

If we throw a die, the six is appear. Then the probability is p and the experiment

… (118 more words) …

Question number: 63

» Probability » Conditional Probability

Appeared in Year: 2011

Essay Question▾

Describe in Detail

Consider the following bivariate p. m. f. of (X, Y):

p (0, 10) = p (0, 20) = 2/18;

p (l, 10) = p (l, 30) = 3/18;

p (1, 20) = p (2, 30) = 4/18;

Obtain the conditional mass functions p (y lx = 2), and p (y lx = 1).

Explanation

The joint probability is given

-

x

Total

-

0

1

2

Y

10

2/18

3/18

0

5/18

20

2/18

4/18

0

6/18

30

0

3/18

4/18

7/18

Total

4/18

10/18

4/18

1

First find the marginal probability of x at x = 0, 1, 2

Equation

p (0) =p

… (75 more words) …

Question number: 64

» Statistical Methods » Bivariate Distributions » Bivariate Normal Distribution

Appeared in Year: 2014

Essay Question▾

Describe in Detail

The marginal distributions of X and Y are given in the following table:

X

1

2

Total

Y

3

4

?

?

?

?

1/4

3/4

Total

1/2

1/2

1

If the co-variance between X and Y is zero, find the cell probabilities and see whether X and Y are independent.

Explanation

Given that covariance between X and Y is zero

Equation

This gives that f (xy) =f (x) f (y)

So, the joint probability density function of x = 1, y = 3 is

Equation

We know that if the covariance is zero, then the conditional distribution is equal to unknown

… (67 more words) …

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