Explanation Definition used: Probability Density Function: “The function f ( x ) is a probability density function for a continuous random variable X if, 1. f ( x ) ≥ 0 for all x ∈ ( − ∞ , ∞ ) . 2. ∫ − ∞ ∞ f ( x ) d x = 1 . 3. The probability that X lies in the interval [ c , d ] is given by P ( c ≤ x ≤ d ) = ∫ c d f ( x ) d x ”. Formula used: “The function f ( x ) is a probability density function for a continuous random variable X . Then, the mean μ = ∫ − ∞ ∞ x f ( x ) d x ”. Calculation: The probability density function is, f ( x ) = { p x p + 1 if x ≥ 1 0 otherwise . The function f is a probability density function of the continuous random variable X only if it satisfies the first two condition of the above definition. The function f ( x ) = 0 if x < 1 and f ( x ) = p x p + 1 if x ≥ 1 . Therefore, f ( x ) ≥ 0 for all x belongs to the interval ( − ∞ , ∞ ) . Compute the value of ∫ − ∞ ∞ f ( x ) d x

Calculus for Business, Economics, Life Sciences, and Social Sciences - Boston U.

15th Edition

ISBN: 9781323047620

Author: Barnett, Ziegler, Byleen

Publisher: Pearson Education

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Chapter 11.4, Problem 65E

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To verify: The function f satisfies the necessary condition for a probability density function, find the mean and explain the restriction on p to make sure that mean exists.

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Chapter 11.4, Problem 64E

Chapter 11.4, Problem 66E

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Chapter 11 Solutions

Calculus for Business, Economics, Life Sciences, and Social Sciences - Boston U.

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 verify: The function f satisfies the necessary condition for a probability density function, find the mean and explain the restriction on p to make sure that mean exists.

Solution Summary: The author explains how the function f satisfies the necessary condition for a probability density function.

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To verify: The function f satisfies the necessary condition for a probability density function, find the mean and explain the restriction on p to make sure that mean exists.

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Chapter 11 Solutions