Explanation Approach: If x is a continuous random variable with probability density function f on [ a , b ] , then the expected value of x is, E ( x ) = μ = ∫ a b x f ( x ) d x The variance of x is, Var ( x ) = ∫ a b ( x − μ ) 2 f ( x ) d x and the standard deviation of x is, σ = Var ( x ) Calculation: Consider the given probability density function, f ( x ) = { 20 x 4 9 if 0 ≤ x ≤ 1 20 9 x 5 if x > 1 The expected value is, E ( x ) = ∫ 0 ∞ x f ( x ) d x = ∫ 0 1 x ( 20 x 4 9 ) d x + lim a → ∞ ∫ 1 a x ( 20 9 x 5 ) d x = ∫ 0 1 ( 20 x 5 9 ) d x + lim a → ∞ ∫ 1 a ( 20 9 x 4 ) d x = 20 x 6 54 | 0 1 + lim a → ∞ ( − 20 27 x 3 ) | 1 a Simplify further, E ( x ) = ( 10 27 − 0 ) + [ lim x → ∞ ( − 20 27 a 3 ) − ( − 20 27 ) ] = 10 9 The variance is,

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Chapter 13.2, Problem 22E

To determine

To find:

The expected value, variance and the standard deviation if they exist for the given probability density function f(x)={20x49 if 0≤x≤1209x5 if x>1.

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Chapter 13.2, Problem 21E

Chapter 13.2, Problem 23E

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The expected value, variance and the standard deviation if they exist for the given probability density function f ( x ) = { 20 x 4 9 if 0 ≤ x ≤ 1 20 9 x 5 if x > 1 .

Solution Summary: The author calculates the expected value, variance, and standard deviation for the given probability density function.

Concept explainers

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To find:

The expected value, variance and the standard deviation if they exist for the given probability density function f(x)={20x49 if 0≤x≤1209x5 if x>1.

Want to see the full answer?

Chapter 13 Solutions

The expected value, variance and the standard deviation if they exist for the given probability density function f ( x ) = { 20 x 4 9 if 0 ≤ x ≤ 1 20 9 x 5 if x &gt; 1 .

Solution Summary: The author calculates the expected value, variance, and standard deviation for the given probability density function.

Concept explainers

Videos

To find: The expected value, variance and the standard deviation if they exist for the given probability density function f(x)={20x49 if 0≤x≤1209x5 if x>1.

Want to see the full answer?

Chapter 13 Solutions

The expected value, variance and the standard deviation if they exist for the given probability density function f ( x ) = { 20 x 4 9 if 0 ≤ x ≤ 1 20 9 x 5 if x > 1 .

To find:

The expected value, variance and the standard deviation if they exist for the given probability density function f(x)={20x49 if 0≤x≤1209x5 if x>1.