# ISS Statistics Paper I (Old Subjective Pattern): Questions 120 - 125 of 165

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## Question number: 120

» Statistical Methods » Correlation Coefficient » Multiple Correlation

Appeared in Year: 2015

Essay Question▾

### Describe in Detail

With 3 variables X 1, X 2 and X 3, it is given that r 13 =0.71, R 1.23 =0.78. Find r 12.3.

### Explanation

We know that

The relationship between the partial and multiple correlations is

## Question number: 121

» Statistical Methods » Correlation Coefficient » Intraclass Correlation

Appeared in Year: 2012

Essay Question▾

### Describe in Detail

For a set of 10 pairs of observations (x i, y j), i = 1 (1) 10, the following calculations are available

Examine at 5 % level of significance if the two variables arc uncorrelated in the population.

### Explanation

First, we calculate the sample correlation coefficient r

for i = 1 to 10

Putting the value in the formula for n = 10

The test hypothesis is

The test statistic is the t-test

The tabulated value is t 0.05, 8… (30 more words) …

## Question number: 122

Appeared in Year: 2009

Essay Question▾

### 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

## Question number: 123

» Probability » Standard Probability Distributions » Lognormal

Appeared in Year: 2014

Essay Question▾

### 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

## Question number: 124

Appeared in Year: 2009

Essay Question▾

### 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… (11 more words) …

## Question number: 125

Appeared in Year: 2009

Essay Question▾

### 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

The integral… (9 more words) …

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