Explanation Definition used: Cumulative distribution Functions: If f is a probability density function and F ( x ) = ∫ − ∞ x f ( t ) d t is the associated cumulative distribution, then 1. F ′ ( x ) = f ( x ) , whenever f is continuous. 2. 0 ≤ F ( x ) ≤ 1 , − ∞ < x < ∞ . 3. F ( x ) is non-decreasing on ( − ∞ , ∞ ) . Proof: Consider the function F ( x ) = { 0.5 − 0.5 x 3 if − 1 ≤ x ≤ 1 0 otherwise . Check the condition 2 from the above mentioned definition for F ( x ) by integrating the derivative of F ( x )

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Chapter 11.2, Problem 46E

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To check: Whether the function F(x)={0.5−0.5x3 if −1≤x≤10 otherwise is a cumulative distribution function or not.

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Chapter 11.2, Problem 45E

Chapter 11.2, Problem 47E

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To check: Whether the function F ( x ) = { 0.5 − 0.5 x 3 if − 1 ≤ x ≤ 1 0 otherwise is a cumulative distribution function or not.

Solution Summary: The author evaluates whether the function F(x) is a cumulative distribution function or not.

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To check: Whether the function F(x)={0.5−0.5x3 if −1≤x≤10 otherwise is a cumulative distribution function or not.

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Chapter 11 Solutions