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Section 15.6 Multivariate probability
Objectives
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Compute probabilities and expectations for jointly distributed variables using double integrals.
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Obtain marginal densities by integrating out one variable from the joint distribution.
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Approximate probabilities numerically with Monte Carlo methods when regions or densities are complex.
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Joint density function of \(X\) and \(Y\text{:}\)
\begin{equation*}
\Prob ((X,Y)\in D)=\iint_{D} f(x,y)\dd{A}
\end{equation*}
-
\begin{align*}
\mu_X \amp = \iint_{D} xf(x,y)\dd{A}\\
\mu_Y \amp = \iint_{D} yf(x,y)\dd{A}
\end{align*}
These are just the center of mass again!
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Marginal probability density functions:
\begin{align*}
f_X(x) \amp = \int_{c}^{d} f(x,y)\dd{y}\\
f_Y(y) \amp = \int_{a}^{b} f(x,y)\dd{x}
\end{align*}
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Could do Monte Carlo integration! Would be good practice still for setting up regions!
