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 01 , 02 , 03 and 04 of an experiment are equally likely. Define A = (01 , 04) , B = (02 , 04) and C = (03 , 04) . 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 = (O1 , O4) , B = (O2 , O4) , C = (O3 , O4)
Their intersection is and Union is AUBUC = (O1 , O2 , O3 , O4)
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…
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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 (µ1 , σ1 2) and N (µ2 , σ2 2) under the assumption that the population means are (i) known and (ii) unknown.
Explanation
Let X1 , X2 , … , Xn and Y1 , Y2 , … Ym are the samples taken from independent N (µ1 , σ1 2) and N (µ2 , σ2 2) .
In this question the hypothesis for testing is
Let s1 2 and s2 2 be the estimates variances of σ1 2 and σ2 2 based on sample sizes n and m.
and
Then
This two chi-square random variable are independent. So, H0 can be test by F-test
where s1 2…
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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 X1 , X2 , … , Xm be i. i. d. random variables with common p. m. f.
obtain the p. m. f. of Sm = X1 + X2 + … . + Xm .
Explanation
Let X1 , X2 , … , Xm 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 Sm = X1 + X2 + … . + Xm , 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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