ISS (Statistical Services) Statistics Paper I (Old Subjective Pattern): Questions 48 - 54 of 165

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Let X have a geometric distribution, then for an two non-negative integers m and n,

Prove it

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Evaluate the following by taking seven ordinates of the integrand:

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Obtain the median and the quartiles of the Cauchy distribution with p. d. f.

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By using Euler-Maclaurin formula, find the sum

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Let X ₁, X ₂, …, X _n be n independently identically distributed random variables following an exponential distribution with mean parameter θ. Obtain the distributions of (i) Maximum (X ₁, X ₂, …, X _n); (ii) Minimum (X ₁, X ₂, …, X _n) and (iii) Median for n = 2m + 1, (m > 0)

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Obtain the characteristic function of X whose pdf is

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Let X be a continuous random variable and have F (x) as the distribution function. If E [X] exists, then show that: