# ISS Statistics Paper I (Old Subjective Pattern): Questions 120 - 124 of 165

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## Question number: 120

» Statistical Methods » Correlation Coefficient » Multiple Correlation

Appeared in Year: 2015

### 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

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## Question number: 121

» Statistical Methods » Correlation Coefficient » Intraclass Correlation

Appeared in Year: 2012

### 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

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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)

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## 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

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## 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)

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