# ISS (Statistical Services) Statistics Paper I (Old Subjective Pattern): Questions 122 - 127 of 165

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## Question number: 122

» Probability » Characteristic Function

Appeared in Year: 2009

### Describe in Detail

Find the density, if its characteristic function is

### Explanation

Rewrite the characteristic function Φ (t)

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

The general form is

Putting the characteristic function

… (159 more words) …

## Question number: 123

» Probability » Standard Probability Distributions » Lognormal

Appeared in Year: 2014

### Describe in Detail

If f _{x} (x) be the probability density function of a lognormal distribution, show that

Where and upper limit is and φ (z) is the distribution function of the standard normal distribution. Hence find E (X) and V (X).

### Explanation

Given that f _{X} (x) has a lognormal distribution, the probability density function is

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

… (187 more words) …

## Question number: 124

» Probability » Conditional Probability

Appeared in Year: 2009

### Describe in Detail

(i) Let X be a random variable such that P [X < 0] = 0 and E [x] exist. Show that P (X ≤ 2E [x] ) ≥ l/2

(ii) Let E [X] = 0 and E [X ^{2}] be finite. Show that P (X ^{2} < 9E [X ^{2}] ) > 8/9

### Explanation

(i) Using Markov inequality, for any random variable and constant a > 0

Here a = 2E (X)

or

(ii) Using Chebyshev inequality, Let X have mean E (X) =µand Var (X) =σ ^{2}, then for any a > 0

In this question a ^{2}

… (132 more words) …

## Question number: 125

» Probability » Standard Probability Distributions » Normal

Appeared in Year: 2009

### Describe in Detail

Suppose that the random variable X has a normal distribution with mean µ and variance σ ^{2}. Let φ be the distribution function of a standard normal variate. Find the density of φ (X-µ/σ). Also find E [φ (X-µ/σ) ].

### Explanation

X has a normal distribution with mean µ and variance σ ^{2}

Let

Let t=

σdt = dx, the upper limit is t = z and the lower limit is same

The density function is

The mean is

… (185 more words) …

## Question number: 126

» Statistical Methods » Correlation Coefficient » Multiple Correlation

Appeared in Year: 2010

### Describe in Detail

Let (X, Y) be jointly distributed with density function

Obtain correlation coefficient between X and Y

### Explanation

First find the marginal distribution of X

find the marginal distribution of Y

The mean of X and Y is

The second moment of X and Y is

… (227 more words) …

## Question number: 127

» Numerical Analysis » Summation Formula » Euler-Maclaurin's

Appeared in Year: 2013

### Describe in Detail

Solve the equation:

Use Euler algorithm and tabulate the solution at x = 0.1,0.2,0.3.

### Explanation

The given differential equation is

with initial condition x _{0} = 0, y _{0} = 0

Using Euler, s method,

where

We obtain y at x = 0.1

Again obtain y at x = 0.2

Similarly y at x = 0.3

… (167 more words) …