# ISS (Statistical Services) Statistics Paper I (Old Subjective Pattern): Questions 144 - 149 of 165

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## Question number: 144

» Probability » Standard Probability Distributions » Normal

Appeared in Year: 2011

### Describe in Detail

Let the joint p. d. f. of (X, Y) be f (x, y) = e ^{-y}, 0 < x < y < ∞.

Obtain the probability P (X + Y ≤ 1).

### Explanation

The Joint p. d. f. is

f (x, y) = e ^{-y}, 0 < x < y < ∞.

Let assume X + Y =U and Y = V, then X = U-V

Using Jacobian technique

The range is 0 < ∞, u ≤ v < ∞

The joint p. d. f

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## Question number: 145

» Statistical Methods » Correlation Coefficient » Multiple Correlation

Appeared in Year: 2011

### Describe in Detail

Describe a test of independence of two normal random variables based on r, the sample correlation coefficient using the t distribution. If n = 10 and r = 0·9, then carry out the test.

### Explanation

Let X _{1}, X _{2}, …, X _{n} and Y _{1}, Y _{2}, …, Y _{n} are samples variables and the correlation coefficient of between this paired samples is denoted by r. If X’s and Y’s are follows independent normal random variables and correlation coefficient is ρ. Then, we test the null hypothesis is that the relationship of random variable X and Y is not linear and the

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## Question number: 146

» Statistical Methods » Correlation Ratio

Appeared in Year: 2009

### Describe in Detail

Define correlation ratio η· Show that

0≤ρ ^{2} ≤η ^{2} ≤1.

### Explanation

In statistics, the correlation ratio is a measure of the relationship between the statistical dispersion within individual categories and the dispersion across the whole population or sample. The measure is defined as the ratio of two standard deviations representing these types of variation.

Suppose a value of X _{i} has n _{i} values of Y say, Y _{i1}

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## Question number: 147

» Probability » Probability Generating Functions

Appeared in Year: 2009

### Describe in Detail

Find the generating function of X whose probability density function is

P [X = r] =pq ^{r-1}, r = 1,2, …, 0 < p < 1, q = 1 - p

### Explanation

The probability generating function is defined as

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## Question number: 148

» Statistical Methods » Non-Parametric Test » Wald-Wolfowitz

Appeared in Year: 2015

### Describe in Detail

Given the random samples

X: 1,5, 7,9, 15,17,21,23

Y: 2,6, 10,12,18,20,26,28,32

From the populations having the distribution function respectively as F _{1} and F _{2}, test the hypothesis

against

at 5 % level of significance by the Wald-Wolfowitz run test. It is given that the critical number of runs at the sample sizes (8,9) at 5 % level is 5.

### Explanation

Wald-Wolfowitz run test uses the data of two random samples, sample 1 is X variables of size n = 8 and sample 2 is Y variables of size m = 9 from two population F _{1} (x) and F _{2} (y) respectively.

The hypothesis under the test is that the null hypothesis is the two population are identical otherwise the alternative is the population is differ.

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## Question number: 149

» Statistical Methods » Correlation Coefficient » Partial Correlation

Appeared in Year: 2010

### Describe in Detail

Find the most likely price in Mumbai corresponding to the price of Rs. 70 at Kolkata from the following:

Kolkata | Mumbai | |

Average price | 65 | 67 |

Standard deviation | 2.5 | 3.5 |

Correlation coefficient between the prices of commodities in the two cities is 0·8.

### Explanation

First we find the regression coefficient β _{YX} where Y is the price at Mumbai and X is the price at Kolkata.

where ρ=0.8, σ _{Y} =3.5, σ _{X} =2.5

The estimated regression equation of Y corresponding to X is

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