%ST402 PMSP slides
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  pdfauthor={Martin Knott},
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\begin{flushright}
ST402\\2003-2004\\ Dr M. Knott, B607\\
\end{flushright}}
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\begin{document}

\foilhead{Lecture 5 - Objective and Outcomes} We look at applications
of our work on multivariate distributions.
\begin{itemize}
\item Laws of large numbers and central limit theorem
\item Transformations of random variables
\item Multivariate normal distribution
\item (Random sums)
\end{itemize}
\foilhead{}After reviewing the notes you should:
\begin{itemize}
\item be able to prove the central limit theorem,
\item know how to work out the density of a transformed random
variables,
\item understand how correlation is represented in multivariate
normal distributions.
\end{itemize}


\foilhead{Types of convergence}
 Let $Y,Y_1,Y_2,\ldots,$ be a collection
of random variables.
\begin{enumerate}
  \item Convergence in distribution: $Y_n \xra{\  D \ } Y$ \newline
   if $P(Y_n \leq y) \lra P(Y \leq y)$ as $n \lra \infty$.
   \foilhead{}
  \item Convergence in probability: $Y_n \xra{\  P \ } Y$ \newline if $P(|Y_n - Y| > \eps)
\lra 0$ as $n \lra \infty$ for all $\eps >0$. \foilhead{}
  \item Convergence almost surely: $Y_n \xra{\  a.s. \ } Y$ \newline
    if $A = \{\omega \in \Omega: Y_n(\omega) \lra Y(\omega) \text{ as } n \lra \infty\} \Rightarrow P(A)
= 1$.
\end{enumerate}

\foilhead{Laws of large numbers}
\begin{enumerate}
 \item The law of large numbers: \newline $\frac{1}{n} S_n \xra{\ D \ }
 \mu$ as $n \lra \infty$.
 \item  Weak law of large numbers: \newline
 $\frac{1}{n} S_n \xra{\ P \ } \mu$ as $n \lra \infty$.
 \foilhead{}
  \item Strong law of large numbers: \newline  $\frac{1}{n} S_n \xra{\ a.s. \ } \mu$
 as $n \lra \infty$.  \item  Central limit theorem:
 $$
 \frac{S_n - n \mu}{\sqrt{n \sigma^2}} \xra{\ D \ } N(0,1) \ \text{
 as } \ n \lra \infty .
 $$
 \end{enumerate}


\foilhead{Bivariate transformations}
 Suppose continuous random variables $(X_1,X_2)$
 transformed to $(Y_1,Y_2)$ by  $T: \R^2 \lra \R^2$.

  The joint density of $Y_1$ and $Y_2$ is given by \small
\begin{align*}
 &f_{Y_1,Y_2}(y_1,y_2)\\
 &= \left\{\begin{array}{ll}
f_{X_1,X_2}(x_1(y_1,y_2), x_2(y_1,y_2)) |J(y_1,y_2)|
 &\text{ if } (y_1,y_2) \text{ is in the range of } $T$, \\
0 &\text{ otherwise},
\end{array}\right.
\end{align*}
\foilhead{}
 where $(x_1(y_1,y_2), x_2(y_1,y_2))' = T^{-1}(y_1,y_2)'$
and
$$
J(y_1,y_2) = \left| \begin{array}{ll}
\dd{x_1}{y_1} &  \dd{x_2}{y_1} \\
\dd{x_1}{y_2} &  \dd{x_2}{y_2}
\end{array} \right| = \dd{x_1}{y_1}   \dd{x_2}{y_2} - \dd{x_2}{y_1}
\dd{x_1}{y_2}.
$$


\foilhead{Standard bivariate normal} if $U$ and $V$ are independent
standard normal random variables, and
 \begin{align*}
  X&=U\\
  Y& = \rho U + \sqrt{1- \rho^2} V
  \end{align*}
   then $Y \sim N(0,1)$ and $\corr(X,Y) =
\rho$. The joint distribution is a standard bivariate normal;
$$
f_{X,Y}(x,y) = \tfrac{1}{2 \pi \sqrt{1 - \rho^2}} e^{-(x^2 - 2
\rho x y + y^2)/(2(1-\rho^2))}.
$$

\foilhead{General bivariate normal} if $X^* = \mu_X + \sigma_X X$ and
$Y^* = \mu_Y + \sigma_Y Y$ then $X^* \sim N(\mu_X, \sigma_X^2)$ and
$Y^* \sim N(\mu_Y, \sigma_Y^2)$ with $\corr(X^*, Y^*) = \rho$ and
$$
f_{X^*, Y^*}(x,y) = \frac{1}{\sigma_X \sigma_Y} f_{X,Y}
(\frac{x-\mu_X}{\sigma_X}, \frac{y - \mu_Y}{\sigma_Y}).
$$

\foilhead{Conditional distribution} of $Y^*$ given $X^*$ is
$$
Y^*|X^* = x \ \sim \ N(\mu_y + \rho
\frac{\sigma_Y}{\sigma_X}(x-\mu_X), \ \sigma_Y^2 (1 - \rho^2)).$$


\foilhead{Random sums} suppose that $X_1, X_2, \ldots$ is a sequence of
independent identically distributed random variables and $S = X_1 +
\ldots + X_N$ where $N$ is a random variable.
\begin{enumerate}
\item Conditional \newline \begin{tabular}[t]{ll}
 expectation: & $E(S|N) = N E(X)$, \\
 variance:    & $\var(S|N) = N \var(X)$, \\
 MGF: & $M_{S|N}(t|N) = [M_X(t)]^N$.
 \end{tabular}
 \foilhead{}
 \item Marginal \newline \begin{tabular}[t]{ll}
  expectation: & $E(S) = E(N) E(X)$, \\
  variance: & $\var(S) = E(N) \var(X)$ \\ & $+ \var(N) [E(X)]^2$, \\
  MGF: & $M_S(t) = M_N(\log M_X(t))$, \\
  CGF: & $K_S(t) = K_N(K_X(t))$.
 \end{tabular}
 \end{enumerate}


\end{document}
