Explanation Given: The probability density function is f ( x ) = 1 9 x e − x 3 over [ 0 , ∞ ) . Concept used: For a probability density function f defined on the interval [ a , b ] associated with a continuous random variable X , 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 ) = 1 9 x e − x 3 over [ 0 , ∞ ) . Substitute 0 for a , 5 for b and 1 9 x e − x 3 for f ( x ) in equation ( 1 ) to obtain the mean value as shown below. μ = ∫ 0 ∞ x ( 1 9 x e − x 3 ) d x = 1 9 ∫ 0 ∞ x ( x e − x 3 ) d x = 1 9 ∫ 0 ∞ ( x 2 e − x 3 ) d x = 1 9 [ [ x 2 e − x 3 − 1 3 ] 0 ∞ − 2 ∫ 0 ∞ x e − x 3 − 1 3 d x ] The above expression can be further simplified as follows. μ = 1 9 [ − 3 ( 0 − 0 ) + 6 ∫ 0 ∞ x e − x 3 d x ] = 1 9 [ 6 [ x e − x 3 − 1 3 ] 0 ∞ + 6 [ e − x 3 − 1 3 ] 0 ∞ ] = 1 9 [ − 18 [ 0 − 0 ] − 18 [ 0 − 1 ] ] = 2 Therefore, the expected value is 2 . Substitute 0 for a , ∞ for b , 2 for μ and 1 9 x e − x 3 for f ( x ) in equation ( 2 ) to obtain the variance as shown below. Var ( X ) = ∫ 0 ∞ ( x − 2 ) 2 ( 1 9 x e − x 3 ) d x = 1 9 ∫ 0 ∞ ( x 2 − 4 x + 4 ) ( x e − x 3 ) d x = 1 9 ∫ 0 ∞ ( x 3 e − x 3 − 4 x 2 e − x 3 + 4 x e − x 3 ) d x = 1 9 [ ∫ 0 ∞ x 3 e − x 3 d x − 4 ∫ 0 ∞ x 2 e − x 3 d x + 4 ∫ 0 ∞ x e − x 3 d x ] The above expression can be further simplified as follows

Applied Calculus for the Managerial, Life, and Social Sciences

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Chapter 10.2, Problem 14E

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The mean, variance and standard deviation of the random variable x associated with the probability density function f(x)=19xe−x3 over [0,∞).

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Chapter 10.2, Problem 13E

Chapter 10.2, Problem 15E

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Applied Calculus for the Managerial, Life, and Social Sciences

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 ) = 1 9 x e − x 3 over [ 0 , ∞ ) .

Solution Summary: The author explains the probability density function f(x) = 19x

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The mean, variance and standard deviation of the random variable x associated with the probability density function f(x)=19xe−x3 over [0,∞).

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Chapter 10 Solutions