# ISS (Statistical Services) Statistics Paper I (Old Subjective Pattern): Questions 42 - 46 of 165

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## Question number: 42

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Appeared in Year: 2013

Essay Question▾

### Describe in Detail

Let X be a random variable with E [X] = 4 and E [X 2] = 20. Use Chebyshev’s inequality to determine a lower bound for the probability P [0 < x < 8].

### Explanation

Let X be a random variable with mean µ and variance σ 2. Then any k > 0, the Chebyshev’s inequality is

or ‎

σ 2 = E [X 2] - (E [X] ) 2 =4

Then, a lower bound for the probability

Using Chebyshev’s inequality

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## Question number: 43

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Appeared in Year: 2014

Essay Question▾

### Describe in Detail

Let P (s) be the probability generating function associated with a non-negative, integer valued random variable X. Show that

### Explanation

To show this we use the left hand side, express the summation term

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## Question number: 44

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Appeared in Year: 2014

Essay Question▾

### Describe in Detail

There are three identical bags U 1, U 2 and U 3. U 1 contains 3 red and 4 black balls; U 2 contains 4 red and 5 black balls; U 3 contains 4 red and 4 black balls. One bag is chosen at random; a ball is drawn at random from the chosen bag and it is found to be red. Find the probability that the first bag is chosen.

### Explanation

Given that there are three identical bags U 1, U 2 and U 3. Then probabilities of selecting a bag are,

Let X be the event of selecting a red ball

Probability of selecting a red ball in U 1 is

Similarly, Probability of selecting a red ball in U 2 is

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## Question number: 45

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Appeared in Year: 2010

Essay Question▾

### Describe in Detail

Show that for any discrete distribution, standard deviation is not less than mean deviation from mean.

### Explanation

Here, we show that standard deviation is greater than mean deviation from mean.

S. D. ≥ mean deviation from mean

(S. D. ) 2 ≥ (mean deviation from mean) 2

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

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## Question number: 46

» Statistical Methods » Non-Parametric Test » Mann-Whitney

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Appeared in Year: 2014

Essay Question▾

### Describe in Detail

Consider two samples as follows:

Sample 1: 1,4, 7,9, 16,17,22,24

Sample 2: 2,6, 10,12,18,20,26,28,32

Test whether the examples have come from the same population by Wilcoxon-Mann-Whitney test. [Given value of Z for α = 0·05 = 1·645, where Z is N (0,1) ]

### Explanation

Let the sample 1 consider the population X and sample 2 consider the population Y. The hypothesis whether two sample comes from same identical population. We test the following null and alternative hypothesis is

In Wilcoxon-Mann-Whitney test, first consider the combined ordered sequence o

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