Probability (ISS (Statistical Services) Statistics Paper I (Old Subjective Pattern)): Questions 53 - 59 of 72

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Question number: 53

» Probability » Standard Probability Distributions » Lognormal

Appeared in Year: 2015

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Let X follow log-normal with parameters µ and σ 2. Find the distribution of Y = aX b, a > 0, -∞ < b < ∞

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Question number: 54

» Probability » Characteristic Function

Appeared in Year: 2009

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Find the density, if its characteristic function is

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Question number: 55

» Probability » Standard Probability Distributions » Lognormal

Appeared in Year: 2014

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If f x (x) be the probability density function of a lognormal distribution, show that

Where and upper limit is and φ (z) is the distribution function of the standard normal distribution. Hence find E (X) and V (X).

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Question number: 56

» Probability » Conditional Probability

Appeared in Year: 2009

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(i) Let X be a random variable such that P [X < 0] = 0 and E [x] exist. Show that P (X ≤ 2E [x] ) ≥ l/2

(ii) Let E [X] = 0 and E [X 2] be finite. Show that P (X 2 < 9E [X 2] ) > 8/9

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Question number: 57

» Probability » Standard Probability Distributions » Normal

Appeared in Year: 2009

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Suppose that the random variable X has a normal distribution with mean µ and variance σ 2. Let φ be the distribution function of a standard normal variate. Find the density of φ (X-µ/σ). Also find E [φ (X-µ/σ) ].

Explanation

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Question number: 58

» Probability » Conditional Probability

Appeared in Year: 2011

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Let (X, Y) have the uniform distribution over the range 0 < y· < x < 1. Obtain the conditional mean and variance of X given Y = y.

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Question number: 59

» Probability » Standard Probability Distributions » Poisson

Appeared in Year: 2015

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Prove that for r = 1, 2, …, n

Explanation

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