# ISS (Statistical Services) Statistics Paper I (Old Subjective Pattern): Questions 139 - 143 of 165

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## Question number: 139

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Appeared in Year: 2013

Essay Question▾

### Describe in Detail

Compute the factorial moments µ (r) and the cumulants k r, r = 1,2, …. . , of Poisson distribution with parameter m.

### Explanation

Let X follows Poisson distribution with parameter m. The density function is

The r th factorial moment of Poisson distribution is

… (121 more words) …

## Question number: 140

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Appeared in Year: 2014

Essay Question▾

### Describe in Detail

X 1, X 2, …, X N are independently, identically distributed random variables. Define S N = X 1 + X 2 + … + X N , where N is a random variable independent of X i, i = 1,2, … N.

Show that the moment generating function (mgt) of S N is

where My (t) is the mgf of a random variable Y. Hence find the mgf of S N when N follows a Poisson distribution with parameter λ. and X i follows an exponential distribution with mean parameter θ, i = 1 to N.

### Explanation

The moment generating function of S N is where S N =X 1 +X 2 +…+X N, X i are i. i. d random variable and N is also a random variable.

X i ‘s are

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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 o

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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

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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

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