# Probability [ISS (Statistical Services) Statistics Paper I (Old Subjective Pattern)]: Questions 1 - 7 of 72

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## Question 1

Appeared in Year: *2011*

### Describe in Detail

Essay▾Show that the square of the one sample t-statistic has the F-distribution. What are its degrees of freedom?

### Explanation

The t-statistic is defined as the ratio of a standard normal variable X ~N (0,1) and the square root of where Y ~ and n is the degree of freedom.

Then we show the square of t-statistic follows F-distribution.

We known that X is standard normal distribution, then X ^{2} follows a chi-square distribution with degree of freedom is 1. Let assume Z =

The jo…

… (71 more words) …

## Question 2

Appeared in Year: *2013*

### Describe in Detail

Essay▾State and prove Lindeberg-Levy Central limit theorem.

### Explanation

Lindeberg-Levy Central limit theorem.

Let Y_{1} , Y_{2} , … , Y_{n} be independent and identically distributed random variables with common mean

E (Y_{i}) = µ and finite positive variance Var (Y_{i}) = σ ^{2} for i = 1,2, … , n

then , the distribution of the um of these random variables

S_{n} = Y_{1} + Y_{2} + … + Y_{n}

tends to the normal distribution with mean n µ and variance nσ…

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## Question 3

Appeared in Year: *2013*

### Describe in Detail

Essay▾If X ~N (0,1) , obtain the distribution of X ^{2} .

### Explanation

X ~N (0,1) . The density function is

Let assume Y = X ^{2}

Let

The limit is also change

Differentiate with respect to y,

So, Y = X ^{2} is follows a chi-square distribution with one degree of freedom.

… (1 more words) …

## Question 4

Appeared in Year: *2014*

### Describe in Detail

Essay▾In a lottery 1000 tickets are sold and the cost of a ticket is if 10. The lottery offers a first prize of if 1,000, two second prizes of if 500 each, and three third prizes of if 100 each. A person purchases a ticket. If X denotes the amount he may get, find E (X) and V (X) .

### Explanation

The probability of purchases a ticket is

First prize amount is 1000

Two second prizes amount is 500 each

Three third prizes amount is 100 each

X denotes the amount he may get

Then, the expected value of X is

The variance of X is

## Question 5

Appeared in Year: *2013*

### Describe in Detail

Essay▾Prove that for among the discrete distributions, the geometric distribution has the lack of memory property.

### Explanation

The property of memory less is that these distributions of ″ time from now to the next period ″ are exactly the same. The property is most easily explained in terms of ″ waiting times.

Suppose X is a discrete random variable whose values is a non-negative. In probability theory, a distribution is said to have lack of memory if

Geometric distribution …

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## Question 6

Appeared in Year: *2011*

### Describe in Detail

Essay▾Prove that the sum of two independent chi-squared random variables is also chi-squared.

### Explanation

Let X and Y are two independent chi-squared random variables with degree of freedom n and m respectively. We proof this by moment generating function. The moment generating function of chi-squared distribution is

Then moment generating function of sum of two random variable (X + Y) is

because X and Y are independent

which is the moment generating fu…

… (50 more words) …

## Question 7

Appeared in Year: *2013*

### Describe in Detail

Essay▾Let P (z) be the probability generating function of the random variable X whose probability distribution is Pr [X = n] = p_{n} , n = 0, I, 2, … Find the generating function of (i) Pr [X > n] ,

(ii) Pr [X < n] and (iii) Pr [X = 2n] .

### Explanation

Let P (z) be the probability generating function of the random variable X whose probability distribution is Pr [X = n] = p_{n} . Then

(i) The generating function of Pr [X > n] is denoted by P (z_{> n})

The generating function is

(ii) The generating function of Pr [X < n] is denoted by P (z_{< n})

The generating function is

(iii) The generating function of Pr [X …

… (7 more words) …