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

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Question number: 8

» Probability » Standard Probability Distributions » Uniform

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Appeared in Year: 2010

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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 b

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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.

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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

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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

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