Explanation Given Information: The probability density function is defined as f ( x ) = 3 x − 3 2 x 2 over the interval [ 0 , 1 ] . 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 ) = 3 x − 3 2 x 2 over the interval [ 0 , 1 ] . 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 1 x ( 3 x − 3 2 x 2 ) d x = ∫ 0 1 ( 3 x 2 − 3 2 x 3 ) d x Use the formula ∫ x n d x = x n + 1 n + 1 and integrate. ∫ 0 1 ( 3 x 2 − 3 2 x 3 ) d x = ( 3 x 3 3 − 3 2 ( x 4 4 ) ) 0 1 = [ x 3 − 3 x 4 8 ] 0 1 Use the Fundamental theorem ∫ a b f ( x ) d x = F ( b ) − F ( a ) and apply the limits. [ x 3 − 3 x 4 8 ] 0 1 = [ ( 1 3 − 3 ( 1 4 ) 8 ) − 0 ] = 1 − 3 8 = 5 8 Thus, the expected value of the provided probability density function is 5 8 . Now calculate the variance. The mean of the probability density function is obtained as 5 8 . 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

Calculus: An Applied Approach (Providence College: MTH 109)

9th Edition

ISBN: 9781285142616

Author: Ron Larson

Publisher: CENGAGE C

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Chapter 9, Problem 14TYS

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To calculate: The expected value, variance and standard deviation of probability density function f(x)=3x−32x2 over the interval [0,1].

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Chapter 9, Problem 13TYS

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Chapter 9 Solutions

Calculus: An Applied Approach (Providence College: MTH 109)

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The expected value, variance and standard deviation of probability density function f ( x ) = 3 x − 3 2 x 2 over the interval [ 0 , 1 ] .

Solution Summary: The author calculates the expected value, variance, and standard deviation of probability density function f(x)=3x-32x

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To calculate: The expected value, variance and standard deviation of probability density function f(x)=3x−32x2 over the interval [0,1].

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Chapter 9 Solutions