# ISS (Statistical Services) Statistics Paper I (Old Subjective Pattern): Questions 38 - 44 of 164

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## Question 38

Appeared in Year: *2009*

### Describe in Detail

Essay▾Given the distribution function

find its probability density function.

### Explanation

The distribution function written in this form

The density function is

where

## Question 39

Appeared in Year: *2013*

### Describe in Detail

Essay▾Show that convergence in probability implies convergence in distribution.

### Explanation

The sequence X_{n} converges to X in probability if for any ε > 0

The sequence X_{n} converges to the distribution of X as n tends to infinity if

For ε > 0,

Hence

… (1)

Note that

… . (2)

From equation (1) and (2)

Taking n tends to infinity

If ε tends to zero,

This shows that

… (5 more words) …

## Question 40

Appeared in Year: *2009*

### Describe in Detail

Essay▾A die is rolled twice. Let A, B, C denote the events respectively that the sum of scores is 6, the sum of scores is 7, and the first score is 4. Are A and C independent? Are B and C independent?

### Explanation

A die is rolled twice, the sample space consists of thirty six outcomes. The sample space is

(1,1) , (1,2) , (1,3) , (1,4) , (1,5) , (1,6)

(2,1) , (2,2) , (2,3) , (2,4) , (2,5) , (2,6)

(3,1) , (3,2) , (3,3) , (3,4) , (3,5) , (3,6)

(4,1) , (4,2) , (4,3) , (4,4) , (4,5) , (4,6)

(5,1) , (5,2) , (5,3) , (5,4) , (5,5) , (5,6)

(6,1) , (6,2) , (6,3) , (6,4) , (…

… (60 more words) …

## Question 41

Appeared in Year: *2013*

### Describe in Detail

Essay▾Find the missing value in the table by assuming a polynomial form for y:

x | 0 | 1 | 2 | 3 | 4 |

y | 1 | 2 | 4 | - | 16 |

### Explanation

we have given that there are four values of y is given, then find the missing value of y at x = 3 using Lagrange՚s interpolation polynomial because we don՚t have linear relationship between x and y. That is, equal intervals on x don՚t result in equal intervals on y and hence we need to use Lagrange interpolation.

The table is,

… (8 more words) …

## Question 42

Appeared in Year: *2013*

### Describe in Detail

Essay▾Let X be a random variable with E [X] = 4 and E [X ^{2}] = 20. Use Chebyshev՚s inequality to determine a lower bound for the probability P [0 < 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

… (8 more words) …

## Question 43

Appeared in Year: *2014*

### Describe in Detail

Essay▾Let P (s) be the probability generating function associated with a non-negative, integer valued random variable X. Show that

### Explanation

To show this we use the left hand side, express the summation term

We know that the probability generating function is defined as

Putting this in the above equation, we prove the following

## Question 44

Appeared in Year: *2014*

### Describe in Detail

Essay▾There are three identical bags U_{1} , U_{2} and U_{3} . U_{1} contains 3 red and 4 black balls; U_{2} contains 4 red and 5 black balls; U_{3} contains 4 red and 4 black balls. One bag is chosen at random; a ball is drawn at random from the chosen bag and it is found to be red. Find the probability that the first bag is chosen.

### Explanation

Given that there are three identical bags U_{1}, U_{2} and U_{3}. Then probabilities of selecting a bag are,

Let X be the event of selecting a red ball

Probability of selecting a red ball in U_{1} is

Similarly, Probability of selecting a red ball in U_{2} is

and probability of selecting a red ball in U_{3} is

Now we use Bayes՚ theorem to find the probability that the red…

… (6 more words) …