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In the development of a new receiver for the transmission of digital information, each received bit is rated as acceptable, suspect, or unacceptable, depending on the quality of the received signal, with probabilities 0.9, 0.08, and 0.02, respectively. Assume that the ratings of each bit are independent. In the rst four bits transmitted, let X denote the number of acceptable bits Y denote the number of suspect bits Then, the distribution of X is binomial with n 4 and p 0.9, and the distribution of Y is binomial with n 4 and p 0.08. However, because only four bits are being rated, the possible values of X and Y are restricted to the points shown in the graph in Fig. 5-1. Although the possible values of X are 0, 1, 2, 3, or 4, if y 3, x 0 or 1. By specifying the probability of each of the points in Fig. 5-1, we specify the joint probability distribution of X and Y. Similarly to an individual random variable, we de ne the range of the random variables (X, Y ) to be the set of points (x, y) in two-dimensional space for which the probability that X x and Y y is positive.

The Number type is an IEEE 754 double-precision 64-bit oating-point number and, as such, it can represent positive and negative numbers, both whole and fractional. Numbers are applicable for almost any kind of measurement and are one of the principal workhorses of ActionScript 3.0. The default value for an unassigned variable of type Number is NaN, which is de ned in the section Edge Cases later in this chapter. Usually Numbers are assigned by setting them equal to a numeric literal, but the Number class also has a constructor, which attempts to convert the object passed in to a number and assigns it to the new Number. Typical usage of a Number is as follows:

4.10 10 5 f X Y (x, y)

5 1.84 10 3 3 4.10 10

var n:Number; trace(n); // NaN n = 12; trace(n); // 12

1.54 10 5 1.38 10 3 3.11 10 2

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Assignment is the typical way to associate a value with a variable; it is rare to see a numeric type s constructor in use.

Figure 5-1 Joint probability distribution of X and Y in Example 5-1.

Use Numbers for any kind of measurement, for values that might contain fractional parts, and for extremely large and small values.

6 3.46 10 4 1.56 10 2 0.2333 1 2.56 10

The int type is a signed 32-bit integer. Despite its lowercase name, it is, like all types, a class. Like Number, int has a constructor that is rarely used. You can create a new int by merely typing a number literal. If you try to assign an int variable a nonwhole number, the fractional part is dropped (rather than rounded). The default value for an unassigned int value is 0. Integer values in Flash Player take up less memory than oating-point numbers, so you can use integer number types for sections of code that need to be maximally optimized, and where fractional parts do not matter. The int type is perfect for counters and is frequently found in for loops:

1.6 10 7

2.88 10 5 1.94 10 3 5.83 10 2 0.6561

for (var i:int = 0; i < 1000; i++)

> ...

If X and Y are discrete random variables, the joint probability distribution of X and Y is a description of the set of points (x, y) in the range of (X, Y ) along with the probability of each point. The joint probability distribution of two random variables is sometimes referred to as the bivariate probability distribution or bivariate distribution of the random variables. One way to describe the joint probability distribution of two discrete random variables is through a joint probability mass function. Also, P(X x and Y y) is usually written as P(X x, Y y). De nition The joint probability mass function of the discrete random variables X and Y, denoted as fXY (x, y), satis es (1) (2) (3) fXY 1x, y2

Another new numeric type found in ActionScript 3.0, uint is an unsigned 32-bit integer. It, too, is a class and has a rarely used constructor. Like int, its default value is 0. In fact, uint is identical to int except in its range of values. The uint type is used for bit math because its value is always a simple binary translation of the value, without special encoding or rules like counting backward in two s complement. It provides unfettered access to all 32 bits of the value in memory. A good example of this is color values, illustrated in Figure 7-1, which pack information for transparency and amounts of red, green, and blue into a single 32-bit eld.

0 1 x, Y y2 (5-1)

FIGURE 7-1

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