Probability (ISS Statistics Paper I (Old Subjective Pattern)): Questions 60 - 63 of 72

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Question number: 60

» Probability » Standard Probability Distributions » Uniform

Appeared in Year: 2011

Essay Question▾

Describe in Detail

Let X have the continuous c. d. f. F (x). Define U = F (x). Show that both - log U arid -log (1 - U) are exponential random variables:

Explanation

Let U = F (x), then the distribution function of G of U is given by

Equation

Since F is non-increasing and its continuous.

G (u) =F (F -1 (u) ) implies G (u) =u

Then the p. d. f is

Equation

Since F is a distribution function takes value… (54 more words) …

Question number: 61

» Probability » Conditional Probability

Appeared in Year: 2012

Essay Question▾

Describe in Detail

You arc given the following information:

(i) In random testing, you test positive for a disease.

(ii) In 5 % of cases, the test shows positive even when the subject does not have the disease.

(iii) In the population at large, one person in 1000 has the disease. What is the conditional probability that you have the disease given that you have been tested positive, assuming that if someone has the disease, he will test positive with probability 1?

Explanation

Let X denotes the test is positive and Y denotes the person has disease. Given that 5 % of cases, the test shows positive even when the subject does not have the disease that is

Equation

where Equation denotes the person has no disease. Also given the test positive with… (41 more words) …

Question number: 62

» 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: 63

» 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) …

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