# Probability (ISS Statistics Paper I (Old Subjective Pattern)): Questions 9 - 13 of 72

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## Question number: 9

» 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

… (177 more words) …

## Question number: 10

» 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… (287 more words) …

## Question number: 11

» 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… (186 more words) …

## Question number: 12

» Probability » Expectation

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… (340 more words) …

## Question number: 13

» 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… (266 more words) …