ISS (Statistical Services) Statistics Paper I (Old Subjective Pattern): Questions 31 - 35 of 165
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Question number: 31
» Probability » Convergence » In Probability
Appeared in Year: 2011
Describe in Detail
Let X 1, X 2, …, X n be a sequence of i. i. d. r. v. s with E (X i) = 0 and V (X i) = 1. Show that the sequence tends to 1 in probability.
Explanation
we have known that is the sample variance of the sequence. The mean is
The convergence in probability is
Using Chebychev’s inequality
Thus, sufficient
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Question number: 32
» Probability » Bayes' Theorem
Appeared in Year: 2012
Describe in Detail
The ith box contains 2i white balls and 6 - 2i black balls, i = 1 (1) 3. A fair die is cast once. 3 balls are taken at random from box 1, box 2 or box 3 according as the die shows up face 1, any of 2 and 3, or any of 4,5 and 6, respectively. Let X denotes the number of white balls drawn. Find E (X).
Explanation
E 1 = Box 1 2 white and 4 black when the fair dice value is x 1 =1
E 2 =Box 2 4 white and 2 black when the fair dice value is x 2 = (2,3)
E 3 =Box 3 6 white when the fair dice value is x 3 = (4,5, 6)
X denotes the number of white balls out of randomly select 3 balls
Probability of selecting a ball in E 1 is
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Question number: 33
» Probability » Standard Probability Distributions » Negative Binomial
Appeared in Year: 2012
Describe in Detail
Items from a large lot are examined one by one until r items with a rare manufacturing defect are found. The proportion of items with this type of defect in the lot is known to be p. Let X denote the number of items needed to be examined. Derive the probability distribution of X, and find E (X).
Explanation
In this question, the sample size is n = x+r given and each trail only two possible outcomes. The probability of defect is same for each trail and trails are independent. The experiment continues until r defectives.
In the given question the number of manufacturing defect are fixed which is r and proportion of item which is defect that is probabilit
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Question number: 34
Appeared in Year: 2011
Describe in Detail
Let (X, Y) have a bivariate distribution with finite moments upto order 2. Show that
(i) E (E (X|Y) ) =E (X)
(ii) V (X) ≥V (X|Y)
Explanation
Let (X, Y) have a bivariate distribution with density function id f (x, y). The conditional expectation is define as
Note that E (X|y) is a function of y. If we allow y to vary over the support of Y, then E (X|y) as a function of the random variable Y. Taking the expectation of this function over the random variable Y,
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Question number: 35
» Probability » Elements of Measure Theory
Appeared in Year: 2013
Describe in Detail
If the probability for n independent events are p 1, p 2, …, p n, then prove that:
(i) none of the events will occur
(ii) at least one event will occur
(iii) at most one event will occur;
Explanation
Let A 1 , A 2, …, A n, n independent events with probabilities p 1, p 2, …, p n.
(i) Find the probability that none of the events will occur
Since events are independent and the probability that one event does not occur is 1-p
So, none of the events will occur
(ii) Find the probability that at
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