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\title{Stat 311-Lecture 13 Sum of random variables}
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%\author{\Large Probability}
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Nov 3, 2005

\begin{enumerate}

\item For random variables $X$ and $Y$ and their joint distribution $f(x,y)$,
the probability of $$P(X\leq Y) = \int \int_{x \leq y} f(x,y) \; dx
dy.$$ If $X$ and $Y$ are independent $Unif(0,1)$, the joint
distribution is a uniform random variable in the unit square.


\item Given the distributions of $X$ and $Y$, we are interested in
investigating the distribution of $X+Y$ for various applications.
Assuming the independence of $X$ and $Y$, the cdf of $X+Y$ is
$$F_{X+Y}(z) = \int_{-\infty}^{\infty} f_Y(y) F_X(z-y) \; dy.$$
$$f_{X+Y}(z) = \int_{-\infty}^{\infty} f_Y(y) f_X(z-y) \; dy.$$


\item We already know that if independent random variables
$X \sim Binomial(n,p)$ and $Y
\sim Binomial(m,p)$, $$X+Y \sim Binomial(n+m,p).$$

\item Problem. If independent random variables $X \sim Unif(0,1)$ and $Y \sim Unif(0,1)$, find the density
of $X+Y$.


\item Problem. The lifetime (in hours) of an electronic component is a random
variable with density function given by
$$f(x)= \lambda e^{-\lambda x}, \; x>0.$$
Two of these components operate independently in a piece of
equipment. Let $X_i$ be the life times of the $i$-th component. Find
the probability density of $X_1+X_2$. {\em Ans.} $f_{X+Y}(z) =
\lambda^2 e^{-\lambda z} z, z \geq 0$. This is $gamma(2, \lambda)$.

\item For $X \sim N(\mu_X, \sigma_X^2), Y \sim N(\mu_Y,
\sigma_Y^2)$, $$\alpha X+\beta Y \sim N(\alpha \mu_X + \beta \mu_Y,
\alpha^2 \sigma_X^2 + \beta^2 \sigma_Y^2).$$

\item For $X \sim Poisson(\lambda_X), Y \sim Poisson(\lambda_Y)$,
$X+Y \sim Poisson(\lambda_X+\lambda_Y)$.
\end{enumerate}


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