Explanation Given Information: The probability density function is defined as f ( x ) = 1 14 over the interval [ 0 , 14 ] . Formula used: For any probability density function f of a continuous random variable x over the interval [ a , b ] , the expected value of x is defined as, μ = E ( x ) = ∫ a b x f ( x ) d x For any probability density function f of a continuous random variable x over the interval [ a , b ] , the variance of x is defined as, V ( x ) = ∫ a b ( x − μ ) 2 f ( x ) d x Here μ is the mean or the expected value of x . For any probability density function f of a continuous random variable x over the interval [ a , b ] , the standard deviation of x is defined as, σ = V ( x ) Calculation: Consider the provided probability density function. f ( x ) = 1 14 over the interval [ 0 , 14 ] . Use the formula E ( x ) = ∫ a b x f ( x ) d x for the provided probability density function to calculate the expected value. So, E ( x ) = ∫ 0 14 x ⋅ 1 14 d x = 1 14 ∫ 0 14 x d x Use the formula ∫ x n d x = x n + 1 n + 1 and integrate. 1 14 ∫ 0 14 x d x = 1 14 ( x 2 2 ) 0 14 = 1 28 [ x 2 ] 0 14 Use the Fundamental theorem ∫ a b f ( x ) d x = F ( b ) − F ( a ) and apply the limits. 1 28 [ x 2 ] 0 14 = 1 28 [ 14 2 − 0 2 ] = 1 28 ( 196 ) = 196 28 = 7 Thus, the expected value of the provided probability density function is 7 . Now calculate the variance. The mean of the probability density function is obtained as 7 . Now use the formula V ( x ) = ∫ a b ( x − μ ) 2 f ( x ) d x for the provided probability density function to calculate the variance

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ISBN: 9780357667231

Author: Larson

Publisher: CENGAGE L

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Chapter 9, Problem 13TYS

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To calculate: The expected value, variance and standard deviation of probability density function f(x)=114 over the interval [0,14].

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Chapter 9, Problem 12TYS

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The expected value, variance and standard deviation of probability density function f ( x ) = 1 14 over the interval [ 0 , 14 ] .

Solution Summary: The author calculates the expected value, variance, and standard deviation of probability density function f(x)=114 over the interval.

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To calculate: The expected value, variance and standard deviation of probability density function f(x)=114 over the interval [0,14].

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Chapter 9 Solutions