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

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Question number: 139

» Probability » Moments and Cumulants

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

Equation

The r th factorial moment of Poisson distribution is

Equation

Equation

Equation

Equation

Equation

Equation

The cumulants of Poisson distribution is

Equation

we known that the moment generating function of X is

Equation

So, the cumulants is… (26 more words) …

Question number: 140

» Probability » Moment Generating Functions

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

Equation

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.

Equation

Equation

Equation

X i ’s are independent identically distributed, so

Equation

Equation

Equation

Equation

In the… (50 more words) …

Question number: 141

» Statistical Methods » Association and Contingency

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

Equation

The variance of the batsmen A is

Equation

Equation

Equation

The coefficient of variation is

Equation

The mean of batsmen B is

Equation

The variance of the batsmen B is

Equation

Equation

Equation

The coefficient of variation is

Equation

Since the average of… (43 more words) …

Question number: 142

» Probability » Standard Probability Distributions » Poisson

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

Equation

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

Equation

because X and Y are independent

Equation<span class="more">… (48 more words) …</span>

Question number: 143

» Probability » Conditional Probability

Appeared in Year: 2009

Essay Question▾

Describe in Detail

The joint density of (X, Y) is

Equation

Find the conditional densities and E [X|Y = 1.5].

Explanation

First find the marginal distribution of X and Y

Equation

Equation

Equation

Equation

The conditional density of X|Y is

Equation

Equation

Equation

The conditional density of Y|X is

Equation

Equation

Equation

Equation

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