Explanation Formula used: The mean, median and standard of a continuous random variable are, (1) Mean μ = ∫ − ∞ ∞ x f ( x ) d x . (2) “If m is a median, then m must satisfy P ( X ≤ m ) = 1 2 is solved by finding cumulative distribution function F ( x ) = ∫ − ∞ x f ( t ) d t ”. (3) Standard deviation σ = V ( X ) where variance V ( X ) = ∫ − ∞ ∞ x 2 f ( x ) d x − μ 2 . Calculation: The probability density function f ( x ) = { p x p + 1 if x ≥ 1 0 otherwise . By the above formula, compute the mean value. μ = ∫ − ∞ 1 ( p x p + 1 ) x d x + ∫ 1 ∞ ( p x p + 1 ) x d x = 0 + p lim b → ∞ ∫ 1 b x − p d x ( ∵ x < 1 ) = p lim b → ∞ [ x − p + 1 − p + 1 ] 1 b = p 1 − p lim b → ∞ [ ( b ) − p + 1 − ( 1 ) − p + 1 ] On further simplification. μ = p 1 − p [ − 1 ] = p p − 1 Thus, the value of mean is μ = p p − 1 , if p > 1 . By using the above formula, compute the median value. F ( x ) = ∫ − ∞ x p t p + 1 d t = ∫ − ∞ 1 p t p + 1 d t + ∫ 1 x p t p + 1 d t = 0 + ∫ 1 x p t p + 1 d t = p [ t − p − p ] 1 x On further simplification. F ( x ) = − [ 1 x p − 1 ] = 1 − 1 x p If m is a median, then m must satisfy P ( X ≤ m ) = 1 2 . F ( m ) = P ( X ≤ m ) 1 − 1 m p = 1 2 − 1 m p = − 1 2 m p = 2 m = ( 2 ) 1 p Thus, the value of median is m = ( 2 ) 1 p

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Author: Barnett

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Chapter 11.4, Problem 68E

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To illustrate: The relationship between mean, median and standard deviation of Pareto random variable using graph and approximate the abscissas of intersection points correct to two decimal places.

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Chapter 11.4, Problem 67E

Chapter 11.4, Problem 69E

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Ch. 11.1 - Evaluate the following, if it converges: 3dx(x1)2.Ch. 11.1 - Prob. 2MP Ch. 11.1 - Prob. 3MP Ch. 11.1 - Prob. 4MP Ch. 11.1 - Prob. 5MP Ch. 11.1 - Prob. 6MP Ch. 11.1 - Prob. 1ED Ch. 11.1 - Prob. 2ED Ch. 11.1 - Prob. 1E Ch. 11.1 - Prob. 2E

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To illustrate: The relationship between mean, median and standard deviation of Pareto random variable using graph and approximate the abscissas of intersection points correct to two decimal places.

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To illustrate: The relationship between mean, median and standard deviation of Pareto random variable using graph and approximate the abscissas of intersection points correct to two decimal places.

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