%ST402 PMSP slides

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\begin{flushright}
ST402\\2003-2004\\ Dr M. Knott, B607\\
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\begin{document}
\foilhead{Lecture 3 - Objective and Outcomes}

Moments provide summary information about distributions.
Appropriate conditioning allows simple solutions to apparently
intractable problem. \small
\begin{itemize}
\item moments and moment generating functions
\item cumulant generating functions
\item functions of random variables
\item joint distributions
\item conditional distributions
\end{itemize}

\foilhead{}
 After reviewing the notes you should:
\begin{itemize}
\item be able to work of mgf and cgf for a variety of different
distributions
\item understand a simple method for generating draws from random
variables
\item be able to get marginal distributions from joint distributions
\end{itemize}


\foilhead{Properties of moments}
\begin{enumerate}
 \item $m_1 = E(X) = \mu$ (mean); $\mu_1 = 0$.
 \item $\mu_2 = E(X^2) -E(X)^2 = \var(X) = \sigma^2$ (variance).
 \item Coefficient of skewness: \newline $\gamma_1 = E[(X- \mu)^3]/ \sigma ^3 = \mu_3 /
\mu_2^{\frac{3}{2}}$.
 \item Coefficient of kurtosis: \newline $\gamma_2 = (E[(X- \mu)^4]/ \sigma^4) - 3 = (\mu_4 / \mu_2^2) - 3$.
\end{enumerate}


\foilhead{Bivariate distribution function}
\begin{enumerate}
 \item \begin{tabular}[t]{l}
$F(-\infty,y) = \lim_{x \rightarrow -\infty} F(x,y) = 0$, \\
$F(x,-\infty) = \lim_{y \rightarrow -\infty} F(x,y) = 0$, \\
$F(\infty,\infty) = \lim_{x \rightarrow \infty,y \rightarrow
\infty} F(x,y) = 1$.
\end{tabular}
 \item $P(x_1 < X \leq x_2, y_1 < Y \leq y_2)$ \newline $= F(x_2, y_2) -
F(x_1, y_2) - [F(x_2,y_1) - F(x_1,y_1)]$.
 \foilhead{}
 \item \begin{tabular}[t]{l}
Right continuous is $x$: \\ $\lim_{h \downarrow 0} F(x+h,y) =
F(x,y)$, \\
Right continuous in $y$: \\ $\lim_{h \downarrow 0} F(x,y+h) =
F(x,y)$.
\end{tabular}
\end{enumerate}

\foilhead{Marginal distribution functions}
\begin{align*}
F_X(x) =& \lim_{y \rightarrow \infty} F_{X,Y}(x,y) =
F_{X,Y}(x,\infty), \\
F_Y(y) =& \lim_{x \rightarrow \infty} F_{X,Y}(x,y) =
F_{X,Y}(\infty,y).
\end{align*}


\foilhead{Discrete bivariate distributions}
\begin{enumerate}
 \item Joint mass function: \newline $f(x,y) = f_{X,Y}(x,y) = P(X=x,Y=y)$.
 \item Probabilities: \newline $P(x_1 < X \leq x_2, y_1 < Y \leq y_2) $
\newline $ =
\sum_{x_1 < x \leq x_2} \sum_{y_1 < y \leq y_2} f(x,y)$.
 \item Marginal mass functions: \newline
\begin{tabular}[t]{l}
$f_X(x) = \sum_y f_{X,Y}(x,y)$, \\
$f_Y(y) = \sum_x f_{X,Y}(x,y)$.
\end{tabular}
\end{enumerate}

\foilhead{Continuous bivariate distributions}
\begin{enumerate}
\item Joint density function: is a positive real-valued function
$f$ such that
$$
F(x,y) = \int_{-\infty}^y \int_{-\infty}^x f(u,v) du dv \quad {\rm
for} \ {\rm each} \ x,y \in \R.
$$
\item Probabilities: \newline $P(x_1 < X \leq x_2, y_1 < Y \leq y_2) =
\int_{y_1}^{y_2} \int_{x_1}^{x_2} f(x,y) dx dy$.
 \foilhead{} \item
Marginal density functions: \newline
\begin{tabular}[t]{l}
$f_X(x) = \int_{-\infty}^{\infty} f_{X,Y}(x,y)dy$, \\
$f_Y(y) = \int_{-\infty}^{\infty} f_{X,Y}(x,y)dx$,
\end{tabular}
\end{enumerate}

\foilhead{Expectation of a function of two variables}
 $ E[g(X,Y)] = $ $
 \left\{\begin{array}{ll}
\sum_y \sum_x g(x,y)
f_{X,Y}(x,y), & \text{discrete case}, \\
\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} g(x,y)
f_{X,Y}(x,y) dx dy, & \text{continuous case}.
\end{array} \right.
$

\foilhead{Discrete conditional distributions}:
\begin{enumerate}
\item Conditional distribution of $Y$ given $X=x$:\\
 $F_{Y|X}(y|x) =
P(Y \leq y|X=x)$.
 \item Conditional mass of $Y$ given $X=x$:
\newline $ f_{Y|X}(y|x) = P(Y=y|X=x) = f_{X,Y}(x,y)/f_X(x)$.
\foilhead{}
\item Relationship between distribution and mass:\\
$F_{Y|X}(y|x) =
\sum_{y_i \leq y} f_{Y|X}(y_i|x)$.
\end{enumerate}

\foilhead{Continuous conditional distributions}
\begin{enumerate}
\item Conditional distribution of $Y$ given $X=x$: \newline
$F_{Y|X}(y|x) = \int_{-\infty}^y (f_{X,Y}(x,v)/f_X(x))dv$. \item
Conditional density of $Y$ given $X=x$: \newline $ f_{Y|X}(y|x) =
f_{X,Y}(x,y)/f_X(x)$.
\end{enumerate}

\foilhead{Conditional, joint and marginal densities} \vS
\begin{enumerate}
\item Conditional mass/density \newline
$ f_{Y|X}(y|x) = \frac{f_{X,Y}(x,y)}{f_{X}(x)}$ \newline $ =
 \left\{\begin{array}{ll}
\frac{f_{X,Y}(x,y)}{\sum_y f_{X,Y}(x,y)}, & \text{discrete case}, \\
\frac{f_{X,Y}(x,y)}{\int_{-\infty}^{\infty} f_{X,Y}(x,y)dy}, &
\text{continuous case}.
\end{array} \right.
$ \item Joint mass/density: \newline $f_{X,Y}(x,y) = f_{Y|X}(y|x)
f_X(x)$.\vS \item Marginal mass/density
$$
f_{Y}(y) =
 \left\{\begin{array}{ll}
\sum_x f_{Y|X}(y|x) f_X(x), & \text{discrete case}, \\
\int_{-\infty}^{\infty} f_{Y|X}(y|x) f_X(x) dx, & \text{continuous
case}.
\end{array} \right.
$$
\item Reverse conditioning
$$
f_{X|Y}(x|y) = \frac{f_{X,Y}(x,y)}{f_X(x)} = \frac{f_X(x)}{f_Y(y)}
f_{Y|X}(y|x).
$$
\end{enumerate}

\end{document}
