機率HW7
===

## 1.
### (a.)
$\int ce^{-3x} dx= 1 \Rightarrow \frac{-c}{3}e^{-3x}|^{\infty}_{0} =1
\Rightarrow 0+\frac{c}{3}=1 \Rightarrow c = 3$
### (b.)
$\int^{0.5}_{0}3e^{-3x}dx =-e^{-3x}|^{0.5}_{0}=-e^{-1.5}+e^0=1-e^{-1.5}\simeq 0.78$

## 2.
### (a.)
$$
f(t)=
\left\{
\begin{eqnarray}
32x^{-3} \qquad x\ge 4\\
0 \qquad otherwise\\
\end{eqnarray}
\right.
$$
### (b.)
$P(X \le 5)=F(5) = 1-\frac{16}{25} = \frac{9}{25}$
$P(X \ge 6) =1-F(6) =1-(1-\frac{16}{36})=\frac{16}{36}=\frac{4}{9}$
$P(5 \le X \le 7)=F(7)-F(5)=(1-\frac{16}{49}) - (1-\frac{16}{25})\simeq 0.313$
$P(1 \le X \le 3.5)=0-0=0$

## 3.
### (a.)
For $Y=X^3$
If $G$ is distribution functions of Y,and $g$ is density functions of Y.
$G(t) = P(Y \le t) =P(X^3 \le t)=P(X \le \sqrt[3]{t}) , -8 < t < 8$
$G(t) = \int^{\sqrt[3]{t}}_{-2} \frac{1}{4}dx=\frac{1}{4}x|^{\sqrt[3]{t}}_{-2} = \frac{1}{4}\sqrt[3]{t}+\frac{1}{2}$

$$
G(t)=
\left\{
\begin{eqnarray}
0 \qquad t \le -8 \\
\frac{1}{4}\sqrt[3]{t}+\frac{1}{2} \qquad -8 < t < 8\\
1 \qquad t \ge 8\\
\end{eqnarray}
\right.
$$

$$
g(t)=G^{\prime}(t)=
\left\{
\begin{eqnarray}
\frac{1}{12} t^{-\frac{2}{3}} \qquad -8 < t < 8\\
0 \qquad otherwise\\
\end{eqnarray}
\right.
$$

### (b.)
For $Z=X^4$
If $H$ is distribution functions of Y,and $h$ is density functions of Z.
$H(t) = P(Z \le t) =P(X^4 \le t)=P(-\sqrt[4]{t} \le X \le \sqrt[4]{t}) , 0 \le t < 16$
$H(t) = \int^{\sqrt[4]{t}}_{-\sqrt[4]{t}} \frac{1}{4}dx=\frac{1}{4}x|^{\sqrt[4]{t}}_{-\sqrt[4]{t}} = \frac{1}{2}\sqrt[4]{t}$

$$
G(t)=
\left\{
\begin{eqnarray}
0 \qquad t < 0 \\
\frac{1}{2}\sqrt[4]{t} \qquad 0 \le t < 16\\
1 \qquad t \ge 16\\
\end{eqnarray}
\right.
$$

$$
h(t)=H^{\prime}(t)=
\left\{
\begin{eqnarray}
\frac{1}{8}t^{\frac{-3}{4}} \qquad 0 \le t < 16\\
0 \qquad otherwise\\
\end{eqnarray}
\right.
$$

## 4.
For $Y=\sqrt[3]{X^2}$
If $G$ is distribution functions of Y,and $g$ is density functions of Y.
$G(t) = P(Y \le t) =P(\sqrt[3]{X^2} \le t)=P(X \le \sqrt{t^{3}}) , 0 \le t < \infty$
$G(t) = \int^{\sqrt[3]{X^2}}_{0} \lambda e^{-\lambda x} dx=-e^{-\lambda x}|^{\sqrt[3]{X^2}}_0 =1-e^{-\lambda\sqrt[3]{X^2}}$

$$
G(t)=
\left\{
\begin{eqnarray}
0 \qquad t < 0 \\
1-e^{-\lambda\sqrt[3]{t^2}} \qquad 0 \le t < \infty\\
\end{eqnarray}
\right.
$$

$$
g(t)=G^{\prime}(t)=
\left\{
\begin{eqnarray}
\frac{3}{2} \lambda t^{0.5}e^{-\lambda t^{1.5}} \qquad 0 \le t < \infty \\
0 \qquad otherwise\\
\end{eqnarray}
\right.
$$

## 5.
$E(e^X) = \int^{\infty}_{0} e^X 3e^{-3X} dx=\int^{\infty}_{0} 3e^{-2X} dx=-\frac{3}{2}e^{-2X}|^{\infty}_{0}=\frac{3}{2}$

## 6.
$Var(X) = E(X^2) - E(X)^2$
$E(X) = \int^{\infty}_{-\infty} X \frac{1}{2} e^{-|X|} dx =0$ (odd function)
$E(X^2) = \int^{\infty}_{-\infty}X^{2}\frac{1}{2}e^{-|X|}dx=\int^{\infty}_{0}X^2e^{-X}dx$

$E(X^2) = \int\frac{1}{2}e^{-|X|}dx=\int X^2e^{-X}dx$
$\int X^2e^{-X}dx=X^{2}(-e^{-X})+2\int Xe^{-X}dx$
$\int Xe^{-X}dx=X(-e^{-X})+\int e^{-X}dx$
$\int e^{-X}dx=-e^{-X}$
$\int Xe^{-X}dx=X(-e^{-X})-e^{-X}$
$\int X^2e^{-X}dx=X^{2}(-e^{-X})+2(X(-e^{-X})-e^{-X})=(-e^{-X})(X^{2}+2X+2)=\frac{X^{2}+2X+2}{-e^{X}}$

$\int^{\infty}_{0}X^2e^{-X}dx=\frac{X^{2}+2X+2}{-e^{X}}|^{\infty}_{0}$

$lim_{a \rightarrow \infty}\frac{X^{2}+2X+2}{-e^{X}}=\frac{\infty}{\infty}$,use L'Hôpital's rule.
$lim_{a \rightarrow \infty}\frac{X^{2}+2X+2}{-e^{X}}=lim_{a \rightarrow \infty}\frac{(X^{2}+2X+2)^{\prime}}{(-e^{X})^{\prime}}=lim_{a \rightarrow \infty}\frac{2X+2}{-e^{X}}=lim_{a \rightarrow \infty}\frac{2}{-e^{X}}=0$
$lim_{a \rightarrow 0}\frac{X^{2}+2X+2}{-e^{-X}}=-2$
$E(X^2)=2$
$Var(X)=E(X^2)-E(X)^2=2-0=2$