# Probability (ISS (Statistical Services) Statistics Paper I (Old Subjective Pattern)): Questions 1 - 8 of 72

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## Question number: 1

» Probability » Distribution Function » Standard Probability Distributions

Appeared in Year: 2011

### Describe in Detail

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,

## Question number: 2

» Probability » Central Limit Theorems

Appeared in Year: 2013

### Describe in Detail

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 ,

## Question number: 3

» Probability » Standard Probability Distributions » Normal

Appeared in Year: 2013

### Describe in Detail

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.

## Question number: 4

» Probability » Expectation

Appeared in Year: 2014

### Describe in Detail

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 number: 5

» Probability » Standard Probability Distributions » Geometric

Appeared in Year: 2013

### Describe in Detail

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

## Question number: 6

» Probability » Standard Probability Distributions » Normal

Appeared in Year: 2011

### Describe in Detail

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

## Question number: 7

» Probability » Probability Generating Functions

Appeared in Year: 2013

### Describe in Detail

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

## Question number: 8

» Probability » Tchebycheffs Inequality

Appeared in Year: 2014

### Describe in Detail

Show that for 40, 000 throws of a balanced coin, the probability is at least 0.99 that the proportion of heads will fall between 0.475 and 0.525.

### Explanation

For balanced coin the probability is p = 0.5. For Bernoulli trails where n = 40, 000, the mean and standard deviation is

Using Chebyshev’s inequality,

The question says that the probability is at least 0.99 that is

So, the number of heads comes between