Probability (ISS (Statistical Services) Statistics Paper I (Old Subjective Pattern)): Questions 20 - 26 of 72

Get 1 year subscription: Access detailed explanations (illustrated with images and videos) to 165 questions. Access all new questions we will add tracking exam-pattern and syllabus changes. View Sample Explanation or View Features.

Rs. 550.00 or

Question number: 20

» Probability » Standard Probability Distributions » Geometric

Appeared in Year: 2010

Essay Question▾

Describe in Detail

Let X have a geometric distribution, then for an two non-negative integers m and n,

Equation

Prove it

Explanation

This proof is a lack of memory property. We known that

Equation

Therefore Equation ………… (1)

The equation is

Equation

Equation

Equation

Equation

Using equation (1)

Equation

Equation

Equation

Since the equation is true. Therefore,

Equation

Question number: 21

» Probability » Standard Probability Distributions » Cauchy

Appeared in Year: 2011

Essay Question▾

Describe in Detail

Obtain the median and the quartiles of the Cauchy distribution with p. d. f.

Equation

Explanation

For find the median and quartile of the Cauchy distribution, q is any quartile. Then, for which value of q, the x value is

Equation

Equation

Equation

Equation

Equation

Equation

For first quartile q = 1/4, then x=-1

For median q = 1/2, then x = 0

For third quartile… (7 more words) …

Question number: 22

» Probability » Characteristic Function

Appeared in Year: 2015

Essay Question▾

Describe in Detail

Obtain the characteristic function of X whose pdf is

Equation

Explanation

The characteristic function of X is

Equation

Let assume x-µ = v, then dx = dv

Equation

Equation

Equation

Assume that Equation, then dv = λ du

Equation

First find the value of this integral, this solution adopts the method of contour integration.

Let assume t > 0, For… (91 more words) …

Question number: 23

» Probability » Expectation

Appeared in Year: 2013

Essay Question▾

Describe in Detail

Let X be a continuous random variable and have F (x) as the distribution function. If E [X] exists, then show that:

Equation

Explanation

We known that in continuous random variable, the mean is

Equation

Equation

We know that F (x) =1-S (x) and f (x) =dF (x) /dx

dF (x) =-dS (x)

Equation

Equation

Equation

Equation

Hence it’s proofed.

Question number: 24

» Probability » Expectation

Appeared in Year: 2012

Essay Question▾

Describe in Detail

Let X 1, X 2, ···, X n be random variables such that

Equation

Also

Equation

Find Equation

Explanation

To find the mean and variance, we use the conditional expectation and conditional variance

Equation

By the given definition of conditional expectation

Equation

Equation

Equation

.

.

.

Equation

Equation

The variance is

Equation

Equation

Equation

Equation

Equation

Equation

Equation

Question number: 25

» Probability » Standard Probability Distributions » Normal

Appeared in Year: 2011

Essay Question▾

Describe in Detail

Let X 1, X 2, …, X n be independent N (0, σ 2) random variables. Obtain the mean and variance of Equation . What is its probability distribution?

Explanation

Let X ~ N (0, σ 2). The density function is

Equation

First we find the distribution of Y = X 2

Equation

Equation

Equation

Let Equation

Equation

The limit is also change

Equation

Differentiate with respect to y,

Equation

Equation

So, Y = X 2 is follows a… (24 more words) …

Question number: 26

» Probability » Characteristic Function

Appeared in Year: 2013

Essay Question▾

Describe in Detail

Find the density function of X whose characteristic function Φ (t) is given by:

Equation

Explanation

Rewrite the characteristic function Φ (t)

Equation

Equation

The density function of X is given by Fourier inversion theorem using the characteristic function,

The general form is

Equation

Putting the characteristic function

Equation

Equation

Equation

Equation

Equation

f Page
Sign In