# ISS (Statistical Services) Statistics Paper I (Old Subjective Pattern): Questions 85 - 89 of 164

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## Question 85

Appeared in Year: *2014*

### Describe in Detail

Essay▾Suppose that all the four outcomes 0_{1} , 0_{2} , 0_{3} and 0_{4} of an experiment are equally likely. Define A = (0_{1} , 0_{4}) , B = (0_{2} , 0_{4}) and C = (0_{3} , 0_{4}) . What can you say about the pairwise independence and mutually independence of the events A, B and C?

### Explanation

Given that there are four outcomes. Define A = (O_{1} , O_{4}) , B = (O_{2} , O_{4}) , C = (O_{3} , O_{4})

Their intersection is and Union is AUBUC = (O_{1} , O_{2} , O_{3} , O_{4})

P (A) = P (B) = P (C) =

P (AUBUC) = 1

The events A, B and C are not pairwise independent because

But the events A and B are pairwise independent because

Similarly the events B and C or A and C are pairw…

… (59 more words) …

## Question 86

Appeared in Year: *2013*

### Describe in Detail

Essay▾Obtain 100 (1 -α) % confidence interval for the ratio of population variances by using two independent random samples from N (µ_{1} , σ_{1} ^{2}) and N (µ_{2} , σ_{2} ^{2}) under the assumption that the population means are (i) known and (ii) unknown.

### Explanation

Let X_{1} , X_{2} , … , X_{n} and Y_{1} , Y_{2} , … Y_{m} are the samples taken from independent N (µ_{1} , σ_{1} ^{2}) and N (µ_{2} , σ_{2} ^{2}) .

In this question the hypothesis for testing is

Let s_{1} ^{2} and s_{2} ^{2} be the estimates variances of σ_{1} ^{2} and σ_{2} ^{2} based on sample sizes n and m.

and

Then

This two chi-square random variable are independent. So, H_{0} can be test by F-test

where s_{1} ^{2}…

… (71 more words) …

## Question 87

Appeared in Year: *2015*

### Describe in Detail

Essay▾Let X be a random variable with E [X] = 3 and E [X ^{2}] = 13. Use Chebyshev՚s inequality to obtain P [-2 < 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 88

Appeared in Year: *2012*

### Describe in Detail

Essay▾Of three independent events A, Band C, A only happens with probability , B only happens with probability and C only happens with probability . Find the probability that at least one of these three events happens.

### Explanation

Given that P (A) = , P (B) = , P (C) =

then probability that at least one event of these three events happens is

The events are independent because only one event happens

## Question 89

Appeared in Year: *2010*

### Describe in Detail

Essay▾Let X_{1} , X_{2} , … , X_{m} be i. i. d. random variables with common p. m. f.

obtain the p. m. f. of S_{m} = X_{1} + X_{2} + … . + X_{m} .

### Explanation

Let X_{1} , X_{2} , … , X_{m} i. i. d. random variables with common p. m. f. is P (X = k) which is a binomail random variables with common parameters n and p respectively. Then, the p. m. f. of S_{m} = X_{1} + X_{2} + … . + X_{m} , sum of random variables are found by moment generating function. The moment generating function of binomial distribution is

So, the moment g…

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