Explanation Given information: E ( X ) = B − ∫ A B F ( x ) d x . Formula used: 1) Let X be a continuous random variable whose possible values lie between A and B , and f ( x ) be the probability density function for X . Then, the expected value (average) of X is the number E ( X ) defined by, E ( X ) = ∫ A B x f ( x ) d x . 2) Integration by parts ∫ f ( x ) ⋅ g ( x ) d x = F ( x ) g ( x ) − ∫ F ( x ) ⋅ g ' ( x ) d x Proof: Consider, F ( x ) is an anti-derivative of f ( x ) and if g ( x ) = x ⇒ g ' ( x ) = 1 . By definition of the expected value, E ( X ) = ∫ A B x f ( x ) d x

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Chapter 12.3, Problem 25E

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To prove: E(X) = B − ∫ABF(x) dx, where F(x) is the cumulative distribution function for X on A ≤ x ≤ B.

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Chapter 12.3, Problem 24E

Chapter 12.3, Problem 26E

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To prove: E ( X ) = B − ∫ A B F ( x ) d x , where F ( x ) is the cumulative distribution function for X on A ≤ x ≤ B .

Solution Summary: The author explains the formula used to prove that E(X) is an anti-derivative of f(x).

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To prove: E(X) = B − ∫ABF(x) dx, where F(x) is the cumulative distribution function for X on A ≤ x ≤ B.

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Chapter 12 Solutions