# Probability-Standard Probability Distributions (ISS Statistics Paper I (Old Subjective Pattern)): Questions 8 - 12 of 22

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## Question number: 8

» Probability » Standard Probability Distributions » Uniform

Appeared in Year: 2010

### Describe in Detail

Let X be a random variable with a continuous distribution function F. Show that F (X) has the uniform distribution on (0, 1).

### Explanation

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

The… (93 more words) …

## Question number: 9

» Probability » Standard Probability Distributions » Cauchy

Appeared in Year: 2012

### Describe in Detail

For the Cauchy distribution given by

where k is a constant to be suitably chosen, derive the expression for the distribution function. Hence obtain a measure of central tendency and a measure of dispersion. What are the points of inflexion of the distribution?

### Explanation

We choose k as constant that gives the integral over the range x for the density function is equal to one.

… (381 more words) …

## Question number: 10

» Probability » Standard Probability Distributions » Normal

Appeared in Year: 2012

### Describe in Detail

12.3 % of the candidates in a public examination score at least 70%, while another 6.3 % score at most 30%. Assuming the underlying distribution to be normal, estimate the percentage of candidates scoring 80 % or more.

### Explanation

Let total marks obtain is 100. Assuming the underlying distribution to be normal, the mean µ and variance σ ^{2}. It is given that

The value of z corresponding to… (173 more words) …

## Question number: 11

» Probability » Standard Probability Distributions » Binomial

Appeared in Year: 2010

### Describe in Detail

Let X _{1}, X _{2}, …, X _{m} be i. i. d. random variables with common p. m. f.

obtain the p. m. f. of S _{m} = X _{1} + X _{2} + …. + X _{m}.

### Explanation

Let X _{1}, X _{2}, …, X _{m} i. i. d. random variables with common p. m. f. is P (X = k) which is a binomail random variables with common parameters n and p respectively. Then, the p. m. f. of S _{m} = X _{1} +… (263 more words) …

## Question number: 12

» Probability » Standard Probability Distributions » Gamma

Appeared in Year: 2011

### Describe in Detail

Let

Show that f (x) is a probability density function. Obtain V (X).

### Explanation

if X is a continuous random variable and f (x) is a continuous function of X, then f (x) is a probability density function if

… (380 more words) …