Lecture 7: Multiple Discrete Random Variables: Expectations, Conditioning, Independence

同理三个事件的PMF $P_{X,Y,Z}(x,y,z)$
$P_X(x)={\sum }_{y,z}{P}_{X,Y,Z}(x,y,z)$

首先应该注意对所有的x，y，z上面的条件都要成立才有独立。
$P_{X|Y}(x|y)=P_X(x)$ 给出新的信息 Y= y并没有对X= x的概率产生影响，这个式子 不能用作定义，因为$P_Y(y)$必须大于0

如果 X 和Y是independent
那么有$P_{X,Y}(x,y)=P_X(x)\ast P_Y(y)$
因为
$$\begin{gathered} E[XY]=\sum _X\sum _{Y}x\ast y\ast P_{X,Y}(x,y) \\ =\sum _X\sum _{Y}x\ast y\ast P_X(x)\ast P_Y(y) \\ =\sum _{X}x\ast P_X(x)\ast \sum _{Y}y\ast P_Y(y) \\ =E[x]\ast E[y] \end{gathered}$$
同理有$E[g(X)h(Y)]=E[g(x)]\ast E[h(y)]$
也可以这么理解
If X and Y are independent, what does it mean? X does not convey any information about Y. If X conveys no information about Y, does X convey information about h(Y)? No. If X tells me nothing about Y, nothing new, it shouldn’t tell me anything about h(Y).

Now, if X tells me nothing about h of h(Y), could g(X) tell me something about h(Y)? No. So the idea is that, if X is unrelated to Y, doesn’t have any useful information, then g(X) could not have any useful information for h(Y). So if X and Y are independent, then g(X) and h(Y) are also independent.

只有当两个random variable是独立的时候方差才能相加，不然会有协方差的项。

如果原始的random variable太复杂的话，可以转化成多个random variable。
There’s lots of problems in probability that you can approach really nicely by breaking up the random variable of interest into a sum of simpler and more manageable random variables. And if you can make it to be a sum of random variables that are just 0’s or 1’s, so much the better. Life is easier. Random variables that take values 0 or 1, we call them indicator variables. They indicate whether an event has occurred or not.
举个例子