Explanation Given information: The median of a probability distribution to be that value a such that ∫ a ∞ p ( x ) d x = ∫ − ∞ a p ( x ) d x = 1 2 Formula used: A probability function p ( z ) is symmetric about the line z = a , then f ( a + z ) = f ( a − z ) , ∀ z . Mean, μ = ∫ − ∞ ∞ x p ( x ) d x Proof: Given condition is ∫ a ∞ p ( x ) d x = ∫ − ∞ a p ( x ) d x = 1 2 . Further simplify the above condition, we get ∫ a ∞ p ( x ) d x + ∫ − ∞ a p ( x ) d x = 1 ∫ − ∞ ∞ p ( x ) d x = 1 The probability function is symmetric about the line x = m . (Given) ⇒ p ( m + x ) = p ( m − x ) , ∀ x Mean is given by μ = ∫ − ∞ ∞ x p ( x ) d x = ∫ − ∞ ∞ ( m + ( x − m ) ) p ( x ) d x = ∫ − ∞ ∞ m p ( x ) d x +

3rd Edition

ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

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Chapter 8.8, Problem 23E

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To show thatm is both mean and median if a probability function is symmetric about the line $x=m.$

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Chapter 8.8, Problem 22E

Chapter 8.8, Problem 24E

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Chapter 8 Solutions

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Ch. 8.1 - Prob. 1PQ Ch. 8.1 - Prob. 2PQ Ch. 8.1 - Prob. 3PQ Ch. 8.1 - Prob. 1E Ch. 8.1 - Prob. 2E Ch. 8.1 - Prob. 3E Ch. 8.1 - Prob. 4E Ch. 8.1 - Prob. 5E Ch. 8.1 - Prob. 6E Ch. 8.1 - Prob. 7E

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To show thatm is both mean and median if a probability function is symmetric about the line x = m .

Solution Summary: The author shows thatm is both mean and median if a probability function is symmetric about the line x=m.

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To show thatm is both mean and median if a probability function is symmetric about the line $x=m.$

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To show thatm is both mean and median if a probability function is symmetric about the line x = m .

Solution Summary: The author shows thatm is both mean and median if a probability function is symmetric about the line x=m.

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To show thatm is both mean and median if a probability function is symmetric about the line x=m.<m:math><m:mrow><m:mi>x</m:mi><m:mo>=</m:mo><m:mi>m</m:mi><m:mo>.</m:mo></m:mrow></m:math>

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Chapter 8 Solutions

To show thatm is both mean and median if a probability function is symmetric about the line $x=m.$