# ISS (Statistical Services) Statistics Paper I (Old Subjective Pattern): Questions 128 - 134 of 165

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## 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

… (128 more words) …

## 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

… (310 more words) …

## 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* −1)! for *r* integer.

… (108 more words) …

## 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

Then find the sign of difference between d _{i} and d _{0}.

-+ –++

… (143 more words) …

## 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

which is always true because this

… (134 more words) …

## Question number: 133

» Statistical Methods » Tests of Significance » Z-Test

Appeared in Year: 2010

### Describe in Detail

How large a sample must be taken in order that the probability will be at least 0·95 that X _{n} will be within 0·5 of µ (µ is unknown and σ = l).

### Explanation

Here, you would know the standard deviation of a population but not know the mean of that population. So, you want to estimate the mean μ to within a given margin of error in a given confidence interval. The required sample size is

Margin of error E = 0.5,

1-α=0.95 →α=0.05

σ=1 and z _{α/2} =1.96

… (41 more words) …

## Question number: 134

» Statistical Methods » Measures of Location

Appeared in Year: 2013

### Describe in Detail

Find the weighted arithmetic mean of the first ‘n’ natural numbers, the weights being the corresponding numbers.

### Explanation

If x _{1}, x _{2}, …, x _{n} are the values having weights w _{1}, w _{2}, …, w _{n} respectively, the weight mean is

Here X is first ‘n’ natural numbers that is 1,2, 3, …, n

… (108 more words) …