Multivariate Distributions

\section{Joint Probability Distributions}

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\begin{defn}
Let $X,Y$ be two random variables defined on the same sample space. Let $A,B$ be the set of possible values for $X,Y$ respectively. The function

$\begin{align*} p(x,y) &= P(X = x, Y = y) \end{align*}$

is called
\begin{enumerate}
\item The Joint Probability Mass Function (jpmf) of $X,Y$ when discrete, with

$\begin{align*} \sum_{x \in A}\sum_{y \in B} p(x,y) = 1 \end{align*}$

\item The Joint Probability Density Function (jpdf) of $X,Y$ when continuous, denoted by $f(x,y)$ with

$\begin{align*} \int_{-\infty}^{\infty}\int_{-\infty}^{\infty} f(x,y)dxdy = 1 \end{align*}$

\end{enumerate}
\end{defn}
}}

\section{Marginal Distributions}

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\begin{defn}[Discrete]
Let $X,Y$ have jpmf $p(x,y)$ , with sets of possible values $A,B$ respectively, then

$\begin{align*} p_X(x) &= \sum_{y \in B} p(x,y) \\ p_Y(y) &= \sum_{x \in A} p(x,y) \end{align*}$

are called the marginal probability mass functions of $X$ and $Y$ .
\end{defn}
}}
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and for the continuous case,

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\begin{defn}[Continuous]
Let $X,Y$ have jpdf $f(x,y)$ , then

$\begin{align*} f_X(x) &= \int_{-\infty}^{\infty} f(x,y)dy \\ f_Y(y) &= \int_{-\infty}^{\infty} f(x,y)dx \end{align*}$

are called the marginal probability density functions of $X$ and $Y$ .
\end{defn}
}}

\section{Expected Values}

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\begin{defn}[Expected Value (Discrete)]
We define the expected value to be

$\begin{align*} E[X] &= \sum_{x \in A} xp_X(x) \\ E[Y] &= \sum_{y \in B} yp_Y(y) \end{align*}$

\end{defn}
}}

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\begin{thm}
Let $p(x,y)$ be the jpmf of $X,Y$ . If $h:\Real ^2 \rightarrow \Real$ , then $h(X,Y)$ is a discrete r.v. with

$\begin{align*} E\left[h(X,Y)\right] &= \sum \sum h(x,y) p(x,y) \end{align*}$

\end{thm}
}}

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\begin{defn}[Expected Value (Continuous)]

$\begin{align*} E[X] &= \int_{-\infty}^{\infty}x f_X(x)dx \\ E[Y] &= \int_{-\infty}^{\infty}y f_Y(y)dy \end{align*}$

\end{defn}
}}

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\hspace{-1cm}\fbox{\parbox{\textwidth + 1cm}{
\begin{thm}
Let $f(x,y)$ be the jpdf of $X,Y$ . If $h:\Real ^2 \rightarrow \Real$ , then $h(X,Y)$ is a discrete r.v. with

$\begin{align*} E\left[h(X,Y)\right] &= \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} h(x,y) p(x,y) \end{align*}$

\end{thm}
}}

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\begin{cor}
By Theorems 3.2 and 3.4 we have

$\begin{align*} E[X + Y] &= E[X] + E[Y] \end{align*}$

\end{cor}
}}

\section{Independent Random Variables}

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\begin{defn}[Expected Value (Continuous)]
Two random variables $X,Y$ are called independent if

$\begin{align*} P(X \in A, Y\in B) &= P(X \in A) P(Y \in B) \\ \shortintertext{ or, } P(X \leq a, Y \leq b) &= P(X \leq a)P(Y \leq b) \end{align*}$

\end{defn}
}}

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\hspace{-1cm}\fbox{\parbox{\textwidth + 1cm}{
\begin{thm}
Let $X,Y$ be two r.v.s. If $F$ is the joint distribution function of $X$ and $Y$ , then $X$ and $Y$ are independent if and only if $\forall t,u \in \Real$

$\begin{align*} F(t,u) &= F_X(t)F_Y(u) \end{align*}$

\end{thm}
}}

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\hspace{-1cm}\fbox{\parbox{\textwidth + 1cm}{
\begin{thm}
Let $X,Y$ be discrete r.v.s with $p(x,y)$ their jpmf, then $X,Y$ are independent if and only if $\forall x,y \in \Real$

$\begin{align*} p(x,y) &= p_X(x)p_Y(y) \end{align*}$

\end{thm}
}}

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\hspace{-1cm}\fbox{\parbox{\textwidth + 1cm}{
\begin{thm}
Let $X,Y$ be independent r.v.s and $g,h : \Real \rightarrow \Real$ , then $g(X)$ and $h(Y)$ are also independent r.v.s.
\end{thm}
}}

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\begin{thm}
Let $X,Y$ be independent r.v.s Then $\forall g,h:\Real \rightarrow \Real$

$\begin{align*} E[g(X)h(Y)] &= E[g(X)] E[h(Y)] \shortintertext{Most importantly} E[XY] &= E[X]E[Y] \end{align*}$

\end{thm}
}}

\section{Conditional Distributions}

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\hspace{-1cm}\fbox{\parbox{\textwidth + 1cm}{
\begin{defn}
The conditional probability mass function of $X$ given that $Y = y$ is

$\begin{align*} p_{X|Y}(x|y) &= P(X = x | Y = x) = \frac{P(X = x, Y = y)}{P(Y = y)} \\ &= \frac{p(x,y)}{p_Y(y)} \end{align*}$

\end{defn}
}}

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\begin{defn}
The conditional expectation of $X$ given that $Y = y$ is

$\begin{align*} E[X | Y = y] = \sum_{x \in A}xP(X = x | Y = y) = \sum_{x \in A} x p_{X|Y}(x|y) \end{align*}$

\end{defn}
}}

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\begin{defn}
The conditional probability density function of $X$ given that $Y = y$ is

$\begin{align*} f_{X|Y}(x|y) &= \frac{f(x,y)}{f_Y(y)} \\ \end{align*}$

\end{defn}
}}

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\hspace{-1cm}\fbox{\parbox{\textwidth + 1cm}{
\begin{defn}
The conditional expectation of $X$ given that $Y = y$ is

$\begin{align*} E[X | Y = y] = \int_{-\infty}^{\infty}xf_{X|Y}(x|y) dx \end{align*}$

\end{defn}
}}