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

Essay Question▾

Describe in Detail

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

Equation

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

Equation

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

Equation

For ε > 0,

Equation

Equation

Hence

Equation ……… (1)

Note that

Equation

Equation …. (2)

From

… (42 more words) …

Question number: 47

» Probability » Distribution Function » Standard Probability Distributions

Appeared in Year: 2010

Essay Question▾

Describe in Detail

Let k > 0 be a constant, and

Equation

Obtain P (X > 0.3).

Explanation

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

Equation

Equation

Equation

Equation

Equation

So, density is

Equation

Then obtain P (X > 0.3)

Equation

Equation

Equation

Equation

Equation

Equation

Equation

Question number: 48

» Probability » Probability of M Events Out of N

Appeared in Year: 2010

Essay Question▾

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),

… (160 more words) …

Question number: 49

» Probability » Tchebycheffs Inequality

Appeared in Year: 2009

Essay Question▾

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

Equation

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

… (11 more words) …

Question number: 50

» Probability » Definitions and Axiomatic Approach

Appeared in Year: 2013

Essay Question▾

Describe in Detail

Let X have the density function,

Equation

(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

Equation

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

… (21 more words) …

Question number: 51

» Probability » Standard Probability Distributions » Geometric

Appeared in Year: 2012

Essay Question▾

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

Equation

This is the probability mass function of geometric distribution

Statement:

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

… (169 more words) …

Question number: 52

» Probability » Standard Probability Distributions » Normal

Appeared in Year: 2014

Essay Question▾

Describe in Detail

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

Equation

where Equation , Equation 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).

Equation

Then

Equation ………. (1)

Let assume that Equation =z, then differentiate Equation

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

Equation (1) can be written as

… (15 more words) …

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