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

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## Question number: 55

» Statistical Methods » Non-Parametric Test » Wilcoxon

Appeared in Year: 2010

### 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.

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## Question number: 56

Appeared in Year: 2012

### 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

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## Question number: 57

» Probability » Standard Probability Distributions » Normal

Appeared in Year: 2011

### 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

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## Question number: 58

» Probability » Characteristic Function

Appeared in Year: 2013

### 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

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## Question number: 59

» Statistical Methods » Tests of Significance » Chi-Square

Appeared in Year: 2010

### 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

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## Question number: 60

» Probability » Standard Probability Distributions » Uniform

Appeared in Year: 2010

### 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 [

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## Question number: 61

» Probability » Standard Probability Distributions » Cauchy

Appeared in Year: 2012

### 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

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