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

Equation

Equation

The relationship between the partial and multiple correlations is

Equation

Equation

Equation

Equation

Equation

Equation

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

Equation

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

Explanation

First, we calculate the sample correlation coefficient r

Equation

for i = 1 to 10

Equation

Putting the value in the formula for n = 10

Equation

Equation

Equation

The test hypothesis is

Equation

The test statistic is the t-test

Equation

Equation

Equation

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

Question number: 122

» Probability » Characteristic Function

Appeared in Year: 2009

Essay Question▾

Describe in Detail

Find the density, if its characteristic function is

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

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

Equation

Where Equation and upper limit is Equation 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

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

Equation

Equation<span class="more">… (60 more words) …</span>

Question number: 124

» Probability » Conditional Probability

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

Equation

Here a = 2E (X)

Equation

Equation

or

Equation

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

Equation

Equation

Equation

In this… (11 more words) …

Question number: 125

» Probability » Standard Probability Distributions » Normal

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

Equation

Let Equation

Equation

Equation

Equation

Let t= Equation

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

Equation

The density function is

Equation

The mean is

Equation

Equation

The integral… (9 more words) …

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