# ISS (Statistical Services) Statistics Paper I (Old Subjective Pattern): Questions 126 - 132 of 165

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## Question number: 126

» Statistical Methods » Correlation Coefficient » Multiple Correlation

Appeared in Year: 2010

### Describe in Detail

Let (X, Y) be jointly distributed with density function

Obtain correlation coefficient between X and Y

### Explanation

First find the marginal distribution of X

find the marginal distribution of Y

The mean of X and Y is

The second moment of X and Y is

The variance of X and Y

## Question number: 127

» Numerical Analysis » Summation Formula » Euler-Maclaurin's

Appeared in Year: 2013

### Describe in Detail

Solve the equation:

Use Euler algorithm and tabulate the solution at x = 0.1, 0.2, 0.3.

### Explanation

The given differential equation is

with initial condition x _{0} = 0, y _{0} = 0

Using Euler, s method,

where

We obtain y at x = 0.1

Again obtain y at x = 0.2

Similarly y at x = 0.3

## Question number: 128

» Probability » Conditional Probability

Appeared in Year: 2011

### Describe in Detail

Let (X, Y) have the uniform distribution over the range 0 < y· < x < 1. Obtain the conditional mean and variance of X given Y = y.

### Explanation

The joint probability density function of (X, Y) is

The marginal distribution of X is

The conditional distribution of X given Y = y

The conditional mean is

The conditional variance is

## Question number: 129

» Numerical Analysis » Interpolation Formulae » Lagrange

Appeared in Year: 2012

### Describe in Detail

Solve the equation f (x) = 0 by using a suitable interpolation formula on the following values:

x | 3 | 4 | 5 | 6 |

f (x) | -2.8 | -1.2 | -0.3 | 1.8 |

### Explanation

To solve the equation f (x) = 0, using Lagrange’s inverse interpolation method

## Question number: 130

» Probability » Standard Probability Distributions » Poisson

Appeared in Year: 2015

### Describe in Detail

Prove that for r = 1, 2, …, n

### Explanation

We known that the L. H. S. is an incomplete gamma function and R. H. S. is a cumulative density function of Poisson distribution.

The incomplete gamma function is

Then

provided that *r* is an integer. Thus recall that Γ (*r*) = (*r*

## Question number: 131

» Statistical Methods » Non-Parametric Test » Sign

Appeared in Year: 2011

### Describe in Detail

Let the temperature before and after administration of aspirin be

Patient | Before | After |

1 | 100.0 | 98.1 |

2 | 102.1 | 97.2 |

3 | 100.6 | 98.6 |

4 | 100.1 | 99.1 |

5 | 101.5 | 97.6 |

6 | 102 | 98.6 |

7 | 99.9 | 98.2 |

8 | 102.7 | 98.1 |

9 | 100.40 | 98.2 |

10 | 100.8 | 97.1 |

Test by the sign test; whether aspirin is effective in reducing temperature. What is the p-value of the calculated statistic?

### Explanation

Let M _{1} and M _{2} is the median temperature of the patients. The hypothesis for test is

Fir this first find the difference of median

Also find the difference of temperature of before and after

d _{i} 2.1 4.9 2.0 1.0 3.9 3.4 1.7 4.6 2.4 3.7

## Question number: 132

» Statistical Methods » Bivariate Distributions » Bivariate Normal Distribution

Appeared in Year: 2010

### Describe in Detail

Show that for discrete distribution β _{2} > 1

### Explanation

We have to prove that

By definition of Kurtosis

Let x _{1}, x _{2}, …, x _{n} are n observations in a set have frequency f _{1}, f _{2, } …, f _{n} and the mean of observation is , then

Assume