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Chapter 1: Problem 3

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

Short Answer

Expert verified
While the exact solutions to these integrals do depend on your ability to solve them correctly, it will be a numerical value for each of the expected values \(E(X)\), \(E[(X+2)^3]\), and \(E[6X-2(X+2)^3]\).

Step by step solution

01

Find E(X)

Firstly, you need to find the expected value \(E(X)\). This involves integrating the product of \(X\) and the pdf, \(f(x)\), over the range \(x\) (-2, 4). Query:\[\int_{-2}^{4} x \cdot f(x) dx = \int_{-2}^{4} x \cdot \frac{x+2}{18} dx\]
02

Find E[(X+2)^3]

Next, the task is to find the expected value \(E[(X+2)^3]\). This involves integrating the product of \((X+2)^3\) and the pdf, \(f(x)\), over the range \(x\) (-2, 4). Query:\[\int_{-2}^{4} (x+2)^3 \cdot f(x) dx = \int_{-2}^{4} (x+2)^3 \cdot \frac{x+2}{18} dx\]
03

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

Finally, you need to find the expected value \(E[6X-2(X+2)^3]\). This involves integrating the product of \([6X-2(X+2)^3]\) and the pdf, \(f(x)\), over the range \(x\) (-2, 4). Query:\[\int_{-2}^{4} [6X-2(X+2)^3] \cdot f(x) dx = \int_{-2}^{4} [6X-2(X+2)^3] \cdot \frac{x+2}{18} dx\]

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