# Probability (ISS (Statistical Services) Statistics Paper I (Old Subjective Pattern)): Questions 46 - 52 of 72

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## Question number: 46

» Probability » Convergence » In Distribution

Appeared in Year: 2014

### Describe in Detail

{X _{n}} is a sequence of independent variables. Show that

where X is a random variable. Is the converse true?

### 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

## Question number: 47

» Probability » Distribution Function » Standard Probability Distributions

Appeared in Year: 2010

### Describe in Detail

Let k > 0 be a constant, and

Obtain P (X > 0.3).

### Explanation

We first find the k value for which the f (x) is purely probability density function that is

So, density is

Then obtain P (X > 0.3)

## Question number: 48

» Probability » Probability of M Events Out of N

Appeared in Year: 2010

### Describe in Detail

A unbaised die is rolled twice. Let A be the event that the first throw shows a number ≤ 2, and B be the event that the second throw shows at least 5. Show that P (AUB) =5/9.

### Explanation

A fair 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),

## Question number: 49

» Probability » Tchebycheffs Inequality

Appeared in Year: 2009

### Describe in Detail

Let X ~ BIN (100, 0.2). Compute P [10 ≤ X ≤ 30].

### Explanation

X ~ BIN (100, 0.2), where n = 100, p = 0.2, q = 0.8

E (X) =np = 20, Var (X) =npq = 16, σ = 4

Let X be a random variable with meanE (X) = µ and variance Var (X) = σ ^{2}. Then any

## Question number: 50

» Probability » Definitions and Axiomatic Approach

Appeared in Year: 2013

### Describe in Detail

Let X have the density function,

(i) Find the constant c.

(ii) Find the distribution function.

(iii) Compute P [X > -1/2].

### Explanation

(I) To find the value of c using this density function, the integral of density is equal to one by probability definition. The range of X is|X| < 1

when X is positive, | X|=X < 1 and X is negative, | X|=-X < 1 that is X >

## Question number: 51

» Probability » Standard Probability Distributions » Geometric

Appeared in Year: 2012

### Describe in Detail

Write down the probability mass function of geometric distribution. State and prove its ‘lack of memory property’. Find also the mean and the variance of the distribution.

### Explanation

If there are number of trails such that the probability of success is p. So, the probability that there are x failures before the first success is

This is the probability mass function of geometric distribution

__Statement: __

Among the all discrete distributions, the geometric distribution has the lack of

## Question number: 52

» Probability » Standard Probability Distributions » Normal

Appeared in Year: 2014

### Describe in Detail

If f _{x} (x) be the probability density function of a N (µ, σ ^{2}) distribution, then show that

where , and φ (x) and Φ (x) are the probability density function and distribution function of the standard normal distribution respectively.

### Explanation

Given that f _{X} (x) be the probability density function follows N (µ, σ ^{2}).

Then

………. (1)

Let assume that =z, then differentiate

The limit is also change, the lower limit is and upper limit is

Equation (1) can be written as