# Probability-Standard Probability Distributions (ISS (Statistical Services) Statistics Paper I (Old Subjective Pattern)): Questions 8 - 13 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 inverse exists. Since F is non-increasing and its continuous

G (u) =F (F ^{-1} (u) ) {F is a distribution of X}

G (u) =u

Then the p. d. f of U = F (x) is given by

Since F is a distribution function takes value in range [0, 1]. Hence

U is a uniform distribution on …

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## 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.

The expression of the distribution function is

Assume

The limit is also change.

U = , L= and put the value of k is

The integral is written as

__Measure of central tendency and __ __measure of dispersion __:

The …

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## 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 an area

0.500 - 0.123 = 0.377

We can write

Similarly, . It is given that

The value of z corresponding to an area

0.500 - 0.063 = 0.437

We can write

From equation (1) …

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## 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} + X _{2} + …. + X _{m}, sum of random variables are found by moment generating function. The moment generating function of binomial distribution is

So, the …

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## 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

Assume but limit is same

This integral is a gamma function

So,

Thus f (x) is a probability density function.

To find the variance, we first find the k ^{th} moment

Since the integrate is th…

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## Question number: 13

» Probability » Standard Probability Distributions » Geometric

Appeared in Year: 2012

### Describe in Detail

Write down the probability mass function of geometric distribution. State and prove its ‘lack of memory property’. Find also the mean and the variance of the distribution.

### Explanation

If there are number of trails such that the probability of success is p. So, the probability that there are x failures before the first success is

This is the probability mass function of geometric distribution

__Statement: __

Among the all discrete distributions, the geometric distribution has the lack of memory property.

__Proof: __

The property…

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