Explanation Given: A continuous random variable with probability density function f ( x ) = 2 9 x ( 3 − x ) is the amount of snowfall in feet in a remote region of Alaska in the month of January. 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 ) = 2 9 x ( 3 − x ) and the value of x varies from 0 to 3 ( 0 ≤ x ≤ 3 ) . Substitute 0 for a , 3 for b and 2 9 x ( 3 − x ) for f ( x ) in equation ( 1 ) to obtain the expected amount of snowfall as shown below: μ = ∫ 0 3 x ( 2 9 x ( 3 − x ) ) d x = 2 9 ∫ 0 3 x 2 ( 3 − x ) d x = 2 9 ∫ 0 3 ( 3 x 2 − x 3 ) d x = 2 9 [ 3 x 3 3 − x 4 4 ] 0 3 The above expression can be further simplified as follows: μ = 2 9 [ 3 ( 3 ) 3 3 − ( 3 ) 4 4 − ( 3 ( 0 ) 3 3 − ( 0 ) 4 4 ) ] = 2 9 [ 6

Applied Calculus for the Managerial, Life, and Social Sciences

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ISBN: 9781305887831

Author: Tan

Publisher: CENGAGE CO

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Chapter 10.2, Problem 19E

To determine

The expected amount of snowfall and also find the variance and standard deviation of the continuous random variable associated with the probability density function f(x)=29x(3−x) over the interval [0,3].

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Chapter 10.2, Problem 18E

Chapter 10.2, Problem 20E

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Applied Calculus for the Managerial, Life, and Social Sciences

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The expected amount of snowfall and also find the variance and standard deviation of the continuous random variable associated with the probability density function f ( x ) = 2 9 x ( 3 − x ) over the interval [ 0 , 3 ] .

Solution Summary: The author explains the probability density function f(x)=29x

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The expected amount of snowfall and also find the variance and standard deviation of the continuous random variable associated with the probability density function f(x)=29x(3−x) over the interval [0,3].

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Chapter 10 Solutions