Explanation Given Information: The provided probability density function is f ( x ) = x 4 + 1 4 in the interval [ 0 , 2 ] . Formula used: For any probability density function f of a continuous random variable x for the interval [ a , b ] , the variance of x is given as, V ( x ) = ∫ a b ( x − u ) 2 f ( x ) d x Where, μ is the mean of x . The alternative formula used to find the variance is given as, V ( x ) = ∫ a b x 2 f ( x ) d x − u 2 The formula used for expected value or mean is given as, μ = ∫ a b x f ( x ) d x Calculation: Consider the probability density function, f ( x ) = x 4 + 1 4 , where x ∈ [ 0 , 2 ] . Now, first calculate the mean μ of x by the use of formula, μ = ∫ a b x f ( x ) d x Substitute f ( x ) = x 4 + 1 4 and a = 0 , b = 2 in the above formula, μ = ∫ 0 2 x [ x 4 + 1 4 ] d x = ∫ 0 2 ( x 2 4 + x 4 ) d x = [ x 3 4 × 3 + x 2 4 × 2 ] 0 2 = [ x 3 12 + x 2 8 ] 0 2 Simplify the above equation, μ = [ x 3 12 + x 2 8 ] 0 2 = ( 2 3 12 + 2 2 8 ) − ( 0 12 + 0 8 ) = 8 12 + 4 8 = 7 6 Consider the formula used for finding the variance, V ( x ) = ∫ a b ( x − u ) 2 f ( x ) d x Substitute f ( x ) = x 4 + 1 4 , mean μ = 7 6 and a = 0 , b = 2 in the above equation, V ( x ) = ∫ 0 2 ( x − 7 6 ) 2 ( x 4 + 1 4

Calculus: An Applied Approach (MindTap Course List)

10th Edition

ISBN: 9781305860919

Author: Ron Larson

Publisher: Cengage Learning

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Chapter 9.3, Problem 15E

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To calculate: The variance of the probability density function f(x)=x4+14 on the interval [0,2] by the use of two methods.

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Chapter 9.3, Problem 14E

Chapter 9.3, Problem 16E

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Theory- Let uf and of be the expectation and variance of a forecast model F for an observations y, and let pf(:) denote the probability density function for the forecast model. Also let G be a different forecast model. Which of the following statements are true? 1. Two forecast scores S(F, y) and S(G, y) are independent 2. The squared error (y – µF)² is a proper score 3. The score (y – µf)²lof is a proper score that improves on the squared error score by taking the variance into account. 4. The logarithmic score – log[pf(y)] is a strictly proper score Enter your answer below as a vector indicating the true statements (c(1), c(1,4) or similar). R Code E Start Over Run Code M Submit Answer 1 2 Next Topic

A restaurant serves three fixed-price dinners costing $12, $15, and $18 including tax and tip. The joint probability mass function p(x,y) of the costs of each member of a dining pair (a host and a guest) is shown in the table below. Cost of guest's dinner (y) $12 $15 $18 Cost of $12 0.05 0.05 0.10 host's $15 0.05 0.10 0.30 dinner (x) $18 0.05 0.20 0.10 Are the dinner costs of the guest and the host independent? O A. Yes, because p(x,y) = p(x)p(y) for all x and y is true O B. No, because p(x,y)= p(X)p(y) for all x and y is true O C. Yes, because p(x,y) = p(x)p(y) for all x and y is false O D. No, because p(x,y)= p(x)p(y) for all x and y is false

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Calculus: An Applied Approach (MindTap Course List)

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The variance of the probability density function f ( x ) = x 4 + 1 4 on the interval [ 0 , 2 ] by the use of two methods.

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To calculate: The variance of the probability density function f(x)=x4+14 on the interval [0,2] by the use of two methods.

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Chapter 9 Solutions