# Probability (ISS (Statistical Services) 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

### 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

Since F is non-increasing and its continuous.

G (u) =F (F ^{-1} (u) ) implies G (u) =u

Then the p. d. f is

Since F is a distribution function takes value

## Question number: 61

» Probability » Conditional Probability

Appeared in Year: 2012

### 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

where denotes the person has no disease. Also given the test positive with

## Question number: 62

» Probability » Moments and Cumulants

Appeared in Year: 2013

### 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

The cumulants of Poisson distribution is

we known that the moment generating function of X is

So, the cumulants is

## Question number: 63

» Probability » Moment Generating Functions

Appeared in Year: 2014

### 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 independent identically distributed, so

In the