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

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Appeared in Year: 2010

Essay Question▾

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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 fun

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Question number: 9

» Probability » Standard Probability Distributions » Cauchy

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Appeared in Year: 2012

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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

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Question number: 10

» Probability » Standard Probability Distributions » Normal

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Appeared in Year: 2012

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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

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Question number: 11

» Probability » Standard Probability Distributions » Binomial

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Appeared in Year: 2010

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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

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Question number: 12

» Probability » Standard Probability Distributions » Gamma

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Appeared in Year: 2011

Essay Question▾

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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

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Question number: 13

» Probability » Standard Probability Distributions » Geometric

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Appeared in Year: 2012

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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 propert

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