Random Variables 隨機變數

For instance, in tossing dice we are often interested in the sum of the two dice and are not really concerned about the separate values of the dice. That is, we may be interested in knowing that the sum is 7 are not be concerned over whether the actual outcome was (1,6) or (2,5) or (3,4) or (4,3) or (5,2) or (6,1). Also, in coin-flipping we may be interested in the total number of heads that occur and not care at all about the actual head-tail sequence that results.

These quantities of interest, or more formally, these real-valued functions defined on the sample sapce, are known as random variables.

Suppose that our experiment consists of tossing 3 fair coins. If we let Ydenote the number of heads appearing, then Y is a random variable taking on one of the values 0,1,2,3 with respective probabilities
P\{Y=0\}&=&\displaystyle P\{(T,T,T)\}=\frac{1}{8} \\ \\
...{8} \\ \\
P\{Y=3\}&=&\displaystyle P\{(H,H,H)\}=\frac{1}{8} \\ \\

Since Y must take on one of the values 0 through 3, we must have

$\displaystyle1=P\left (\bigcup_{i=0}^3\{Y=i\}\right )=\sum_{i=0}^3P\{Y=i\}$
which, of course, is in accord with the above probabilities.          $\rule[0.02em]{1.0mm}{1.5mm}$