# ISS (Statistical Services) Statistics Paper I (Old Subjective Pattern): Questions 30 - 34 of 165

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## Question number: 30

» Numerical Analysis » Interpolation Formulae » Lagrange

Appeared in Year: 2013

### Describe in Detail

Use Simpson’s one-third rule to estimate approximately the area of the cross section of a river 80 feet wide, the depth d (in feet) at a distance x from one bank being given by the following table. :

x | 0 | 10 | 20 | 30 | 40 | 50 | 60 | 70 | 80 |

d | 0 | 4 | 7 | 9 | 12 | 15 | 14 | 8 | 3 |

### Explanation

T he one-third rule of the integrand using Simpson’s rule is

The area of the cross section of a river 80 feet wide is

Here n = 8, b = 80, a = 0, h= (b-a) /n = 10

… (99 more words) …

## 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

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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 random

… (57 more words) …

## 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 proba

… (51 more words) …

## 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.

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