# ISS (Statistical Services) Statistics Paper I (Old Subjective Pattern): Questions 141 - 146 of 165

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## Question number: 141

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Appeared in Year: 2014

Essay Question▾

### Describe in Detail

The runs scored by two batsmen A and B in five cricket matches were as follows:

Batsmen A: 50 60 100 70 20

Batsmen B: 120 100 30 20 40

Discuss the consistency and efficiency of the batsmen.

### Explanation

The mean of batsmen A is

The variance of the batsmen A is

The coefficient of variation is

The mean of batsmen B is

The v

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## Question number: 142

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Appeared in Year: 2011

Essay Question▾

### Describe in Detail

Show that the sum of two independent Poisson random variables with parameters λ and µ respectively is a Poisson random variable with parameter λ+µ.

### Explanation

Let X and Y are independent Poisson random variables with parameters λ and µ respectively. We proof this by moment generating function. The moment generating function of Poisson distribution is

So, the sum of X and Y moment generating function is

because X and Y are independent

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## Question number: 143

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Appeared in Year: 2009

Essay Question▾

### Describe in Detail

The joint density of (X, Y) is

Find the conditional densities and E [X|Y = 1·5].

### Explanation

First find the marginal distribution of X and Y

The conditional density of X|Y is

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## Question number: 144

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Appeared in Year: 2011

Essay Question▾

### 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. of U and V is

The marginal distribution of u is

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## Question number: 145

» Statistical Methods » Correlation Coefficient » Multiple Correlation

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Appeared in Year: 2011

Essay Question▾

### 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 a

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## Question number: 146

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Appeared in Year: 2009

Essay Question▾

### 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, Y i

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