Chapter 6

Suppose a random variable \(Z\) has CDF $$ F(z)= \begin{cases}0, & \text { if } z<0 \\ z^{2} / 9, & \text { if } 0 \leq z \leq 3 \\ 1, & \text { if } z>3\end{cases} $$ Find each of the following: (a) \(P(Z>1)\) (b) \(P(1

Find the volume of the solid generated by revolving the region bounded by the curve \(y^{2}=x^{3}\), the line \(y=8\), and the \(y\)-axis: (a) about the line \(x=4\); (b) about the line \(y=8\).

Find the area of the region in the first quadrant below \(y=e^{-x}\) above \(y=\frac{1}{2}\).

The expected value of a function \(g(X)\) of a continuous random variable \(X\) having PDF \(f(x)\) is defined to be \(E[g(X)]=\) \(\int_{A}^{B} g(x) f(x) d x\). If the PDF of \(X\) is \(f(x)=\frac{15}{512} x^{2}(4-x)^{2}\), \(0 \leq x \leq 4\), find \(E(X)\) and \(E\left(X^{2}\right)\).

Find the area of the region trapped between \(y=x e^{-x^{2}}\) and \(y=x / 4\). Hint: There are two separate regions.

The circle \(x=a \cos t, y=a \sin t, 0 \leq t \leq 2 \pi\), is revolved about the line \(x=b, 0

A continuous random variable \(X\) has \(\mathrm{PDF} f(x)=\) \(\frac{3}{256} x(8-x), 0 \leq x \leq 8\). Find \(E\left(X^{2}\right)\) and \(E\left(X^{3}\right)\).

Use the Parabolic Rule with \(n=8\) to approximate the area of the region trapped between \(y=1-e^{-x^{2}}\) and \(y=e^{-x^{2}}\).

Using the same axes, draw the graphs of \(y=x^{n}\) on \([0,1]\) for \(n=1,2,4,10\), and 100 . Find the length of each of these curves. Guess at the length when \(n=10,000\).

Find the area of the region trapped between \(y=\sin x\) and \(y=\frac{1}{2}, 0 \leq x \leq 17 \pi / 6\).