# Probability (ISS Statistics Paper I (Old Subjective Pattern)): Questions 57 - 62 of 72

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## Question number: 57

» Probability » Standard Probability Distributions » Normal

Appeared in Year: 2009

### Describe in Detail

Suppose that the random variable X has a normal distribution with mean µ and variance σ ^{2}. Let φ be the distribution function of a standard normal variate. Find the density of φ (X-µ/σ). Also find E [φ (X-µ/σ) ].

### Explanation

X has a normal distribution with mean µ and variance σ ^{2}

Let … (166 more words) …

## Question number: 58

» Probability » Conditional Probability

Appeared in Year: 2011

### Describe in Detail

Let (X, Y) have the uniform distribution over the range 0 < y· < x < 1. Obtain the conditional mean and variance of X given Y = y.

### Explanation

The joint probability density function of (X, Y) is

The marginal distribution of X is

The conditional distribution of X given… (101 more words) …

## Question number: 59

» Probability » Standard Probability Distributions » Poisson

Appeared in Year: 2015

### Describe in Detail

Prove that for r = 1, 2, …, n

### Explanation

We known that the L. H. S. is an incomplete gamma function and R. H. S. is a cumulative density function of Poisson distribution.

… (271 more words) …

## 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… (284 more words) …

## 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… (90 more words) …

## 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

… (306 more words) …