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Chapter 1: Problem 2

Let \(X\) have the pdf \(f(x)=(x+2) / 18,-2

Short Answer

Expert verified
Three Expected Values are \(E(X)\), \(E[(X+2)^3]\), \(E[6X - 2(X+2)^3]\), the exact values depend on the performance of the integrals and might need further simplification before they can be obtained.

Step by step solution

01

Calculate E(X)

Find the expected value of X using \(E(X) = \int_{-\infty}^{\infty} x f(x) dx\). For the given pdf, this correspond to integrals like \(\int_{-2}^{4} x \cdot (x+2) / 18 dx\). Use the power rule from calculus to perform the integration.
02

Calculate E[(X+2)^3]

Use the same rule to find \(\int_{-2}^{4} (x+2)^3 \cdot (x+2) / 18 dx\). Note that you have to perform the integral of a polynomial here, which again involves using the power rule.
03

Calculate E[6X - 2(X+2)^3]

Perform the same formula again to find \(\int_{-2}^{4} (6X - 2(X+2)^3) \cdot(x+2) / 18 dx\). This will be a mix between steps 1 and 2.
04

Simplify the obtained values

Once the integrals are performed, simplify the terms and obtain the results.

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Most popular questions from this chapter

Let \(X\) have the pdf \(f(x)=4 x^{3}, 0

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