chapter5_exercise.lyx

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\begin_body

\begin_layout Section*
5.6 Normal Distribution
\end_layout

\begin_layout Standard
17.
 Suppose 
\begin_inset Formula $\log(X)\sim N(\mu,\sigma)$
\end_inset

 .
 Find Distribution of 
\begin_inset Formula $X$
\end_inset

.
 
\begin_inset Formula $Y=r(X)$
\end_inset


\end_layout

\begin_layout Standard

\series bold
The Change of Variables Formula 
\end_layout

\begin_layout Standard
Suppose that r is strictly increasing on 
\begin_inset Formula $S$
\end_inset

 .
 for 
\begin_inset Formula $y\in T$
\end_inset

,
\end_layout

\begin_layout Standard
a.
 
\begin_inset Formula $G(y)=F[r^{-1}(y)]$
\end_inset


\end_layout

\begin_layout Standard
b.
 
\begin_inset Formula $g(y)=f[r^{-1}(y)]\frac{d}{dy}r^{-1}(y)$
\end_inset

.
\end_layout

\begin_layout Standard
\begin_inset Formula $y=r^{-1}(x)=log(x)$
\end_inset

, so 
\begin_inset Formula $x=r(y)=e^{y}$
\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula $f(x)=g(r^{-1}(x))\frac{dr^{-1}(x)}{dx}=....$
\end_inset


\end_layout

\begin_layout Standard
18.
 
\series bold
Cauchy Distribution 
\begin_inset Formula $f(x)=\frac{1}{\pi(1+x^{2})}$
\end_inset

 
\series default
for 
\begin_inset Formula $-\infty<x<\infty$
\end_inset

 
\end_layout

\begin_layout Subsection*
Gamma Distribution
\end_layout

\begin_layout Standard
2.
 Quantile function of the exponential distribution with parameter 
\begin_inset Formula $\beta$
\end_inset

: 
\begin_inset Formula $\Phi(t)=\beta\int_{0}^{t}e^{-\beta t}dx=1-e^{-\beta t}=p$
\end_inset

, so 
\begin_inset Formula $t=\Phi^{-1}(p)=\frac{-\log(1-p)}{\beta}$
\end_inset

.
\end_layout

\begin_layout Standard
6.
 
\begin_inset Formula $X_{1},X_{2},...,X_{n}$
\end_inset

 form a random sample of size 
\begin_inset Formula $n$
\end_inset

 from exponential distribution with parameter 
\begin_inset Formula $\beta$
\end_inset

.
 
\begin_inset Formula $\bar{X_{n}}\sim\Gamma(\alpha,n\beta)$
\end_inset

 (Using 
\begin_inset Formula $\psi$
\end_inset

 to prove that)
\end_layout

\begin_layout Subsection*
5.9 Multinomial distribution
\end_layout

\begin_layout Standard
7.
 
\begin_inset Formula $X_{i}$
\end_inset

 is Poisson Distribution, 
\begin_inset Formula $X_{i}\sim\frac{\lambda_{i}e^{-\lambda_{i}}\lambda_{i}^{x}}{x!}$
\end_inset

 
\begin_inset Formula $E[X_{i}]=\lambda_{i}$
\end_inset


\end_layout

\begin_layout Standard
Because 
\begin_inset Formula $X$
\end_inset

 is multinomial distribution, 
\begin_inset Formula $E[X_{i}]=np_{i}$
\end_inset

.
 So 
\begin_inset Formula $p_{i}=\frac{\lambda_{i}}{n}$
\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula $n=\sum_{i}X_{i}\Rightarrow E[n]=\sum_{i}E[X_{i}]\Rightarrow n=\sum_{i}\lambda_{i}$
\end_inset

.
 So 
\begin_inset Formula $p_{i}=\frac{\lambda_{i}}{\sum_{i}\lambda_{i}}$
\end_inset


\end_layout

\end_body
\end_document