Advanced Topics in Statistics: Day 2

Time for some more probability review and some new material on classes of distributions. Chapter numbers refer to Casella and Berger (2002).

More review of probability:
2.2:  Expected Values: *Very Important*
$$E\left[g\left(X\right)\right] = \int_{-\infty}^{\infty}g\left(x\right)f_{X}\left(x\right)\,\,\mathrm{d}x,$$ when $X$ is a continuous random variable.
$$E\left[g\left(X\right)\right] = \sum_{x}g\left(x\right)f_{X}\left(x\right),$$ when $X$ is a discrete random variable.
Properties of $E\left[\cdot\right]$:

• $E\left[a\right] = a, a\in\mathbb{R}$
• $E\left[ag_{1}\left(X_{1}\right) + bg_{2}\left(X_{2}\right) + c\right] = aE\left[g_{1}\left(X_{1}\right)\right] + bE\left[g_{2}\left(X_{2}\right)\right] + c$
• If $g_{1}\left(x\right)\le g_{2}\left(x\right)\le \cdots \le g_{n}\left(x\right), \forall x,$ then $E\left[g_{1}\left(x\right)\right]\le E\left[g_{2}\left(x\right)\right]\le\cdots\le E\left[g_{n}\left(x\right)\right]$.

Remark I
In measure theoretic notation the generalized expected value is, $$E\left[x\right] = \int_{\omega\in\Omega}X\left(\omega\right)\,\,\mathrm{d}P\,\left(\omega\right).$$ I'm not sure exactly what this means; I'll have to come back to it later.

Remark II
What about interchanging sums and integrals?$\newcommand{\?}{\stackrel{?}{=}}$
$$\begin{eqnarray*}
\int\int f\left(x,y\right)\,\,\mathrm{d}x\,\mathrm{d}y&\?&\int\int f\left(x,y\right)\,\,\mathrm{d}y\,\,\mathrm{d}x\\
\sum_{j}\sum_{k}a_{jk}&\?&\sum_{k}\sum_{j}a_{jk}\\
\sum_{j}\left(\int f_{j}\left(x\right)\,\,\mathrm{d}x\right) &\?& \int\left(\sum_{j}f_{j}\left(x\right)\right)\,\,\mathrm{d}x
\end{eqnarray*}$$
For example, $a \ge 0, f \ge 0, f_{j} \ge 0$ then it works. Or, take the absolute values inside the sum (integral), and show that one of the sides is finite.

More to follow on classes of distributions.