Explanation Given: The probability density function is f ( x ) = 3.5 x − 4.5 over the interval [ 1 , ∞ ) . Concept used: The probability density function f defined on the interval [ a , b ] associated with a continuous random variable X then the expected value μ of X is written as follows: μ = ∫ a b x f ( x ) d x ...... (1) The variance of X is written as follows: Var ( X ) = ∫ a b ( x − μ ) 2 f ( x ) d x ...... (2) The standard deviation of X is written as follows: σ = Var ( X ) ...... (3) Calculation: The probability density function is given as f ( x ) = 3.5 x − 4.5 over the interval [ 1 , ∞ ) . To obtain the mean value, substitute 1 for a , ∞ for b and 3.5 x − 4.5 for f ( x ) in equation ( 1 ) as shown below: μ = ∫ 1 ∞ x ( 3.5 x − 4.5 ) d x = 3.5 ∫ 1 ∞ ( x − 3.5 ) d x = 3.5 [ − 1 2.5 x 2.5 ] 1 ∞ The above expression can be further simplified as follows: μ = 3.5 [ − 1 2.5 ( ∞ ) 2.5 − ( − 1 2.5 ( 1 ) 2.5 ) ] = 3.5 ( 0 + 1 2.5 ) = 1.4 Therefore, the mean of X is 1

Applied Calculus for the Managerial, Life, and Social Sciences (MindTap Course List)

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Chapter 10.2, Problem 12E

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The mean, variance and standard deviation of the random variable X associated with the probability density function f(x)=3.5x−4.5 over the interval [1,∞).

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Chapter 10.2, Problem 11E

Chapter 10.2, Problem 13E

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Applied Calculus for the Managerial, Life, and Social Sciences (MindTap Course List)

Ch. 10.1 - Define the following terms in your own words: a....Ch. 10.1 - Prob. 2CQ Ch. 10.1 - Prob. 3CQ Ch. 10.1 - Prob. 1E Ch. 10.1 - Prob. 2E Ch. 10.1 - Prob. 3E Ch. 10.1 - Prob. 4E Ch. 10.1 - Prob. 5E Ch. 10.1 - Prob. 6E Ch. 10.1 - Prob. 7E

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The mean, variance and standard deviation of the random variable X associated with the probability density function f ( x ) = 3.5 x − 4.5 over the interval [ 1 , ∞ ) .

Solution Summary: The author calculates the probability density function f(x)=3.5x-4.5 over the interval

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The mean, variance and standard deviation of the random variable X associated with the probability density function f(x)=3.5x−4.5 over the interval [1,∞).

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Chapter 10 Solutions