# 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

## 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 condition is that convergence in probability to 1 is that .

## 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… (59 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… (70 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… (66 more words) …