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7-61. A lot consists of N transistors, and of these M (M N) are defective. We randomly select two transistors without replacement from this lot and determine whether they are defective or nondefective. The random variable 1, if the ith transistor is nondefective i 0, if the ith transistor is defective Thus, consistency is a large-sample property, describing the limiting behavior of n as n tends to in nity. It is usually dif cult to prove consistency using the above de nition, although it can be done from other approaches. To illustrate, show that X is a consistent estimator of (when 2 ) by using Chebyshev s inequality. See Section 5-10 (CD Only). 7-65. Order Statistics. Let X1, X2, p , Xn be a random sample of size n from X, a random variable having distribution function F(x). Rank the elements in order of increasing numerical magnitude, resulting in X(1), X(2), p , X(n), where X(1) is the smallest sample element (X(1) min{X1, X2, p , Xn}) and X(n) is the largest sample element (X(n) max{X1, X2, p , Xn}). X(i) is called the ith order statistic. Often the distribution of some of the order statistics is of interest, particularly the minimum and maximum sample values. X(1) and X(n), respectively. Prove that the cumulative distribution functions of these two order statistics, denoted respectively by FX112 1t2 and FX1n2 1t2 are FX112 1t2 FX1n2 1t2 1 3 F1t2 4 31

Although the pixel is the basic unit of measurement in Flash, space is continuous, and any measurement can be subdivided. Any display object can be placed at any fraction of a pixel. The rendering engine in Flash Player decides how to rasterize fractional pixels depending on certain quality settings. It s not a bad idea to ensure nonrotated bitmaps exist on whole pixels if you want them to remain crisp.

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Once you place this display object inside another, its space is subject to whatever transformations you apply to it. Perhaps you have a clip called tri, which has a triangle with its top-left corner at (2, 2). This tri clip is shown in Figure 14-2. Maybe you add this clip to the stage. Move the clip 5 pixels to the right, for instance, and what once appeared at (2, 2) is at (5, 2), relative to the stage s coordinate system. In tri s local coordinate system, nothing has moved or changed, nor will it without your direct intervention. Try to think of these as two independent spaces: tri s coordinate system, and the stage s coordinate system. Now, the coordinate system that the stage de nes is not as exible. If you forget for a moment the stage scale modes and alignment that you can do when resizing the Flash Player embed or window, the stage is really xed. You always have this ultimate frame of reference, the global coordinate system. So when you re thinking about where something is on-screen, remember to ask in relationship to what In most cases, it means in relation to the parent coordinate system. In Figure 14-2, you d say that the triangle inside the display object tri is at (2, 2), and the display object tri is at (5, 0). You ll see more about moving between frames of reference when you look at the geometry classes later in this chapter. And 34, Geometric and Color Transformations, goes deeper into how coordinate spaces may be transformed.

Determine the joint probability function for X1 and X2. What are the marginal probability functions for X1 and X2 Are X1 and X2 independent random variables 7-62. When the sample standard deviation is based on a random sample of size n from a normal population, it can be shown that S is a biased estimator for . Speci cally, E1S 2 12 1n 12 1n 22 3 1n 12 24

The display list is mutable it can be changed. In ActionScript 3.0, you have full access to every object on the display list. Not only can you add in display objects, you can rearrange them, and of course remove them. Now s a good time to look at some code and see how simple these operations are in ActionScript.

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