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A random variable is sometimes de ned as a function of one or more random variables. The CD material presents methods to determine the distributions of general functions of random variables. Furthermore, moment-generating functions are introduced on the CD

float2x2 * float2x2 -> float2x2 //multiplies 2x2 matrix by 2x2 matrix, yielding 2x2 matrix float2 * float2x2 -> float2 //multiplies 1x2 matrix by 2x2 matrix, yielding 1x2 matrix (row vector) float2x2 * float2 -> float2 //multiplies 2x2 matrix by 2x1 matrix, yielding 2x1 matrix (column vector) float * float2x2 -> float2x2 //multiplies scalar by 2x2 matrix, yielding 2x2 matrix

and used to determine the distribution of a sum of random variables. In this section, results for linear functions are highlighted because of their importance in the remainder of the book. References are made to the CD material as needed. For example, if the random variables X1 and X2 denote the length and width, respectively, of a manufactured part, Y 2X1 2X2 is a random variable that represents the perimeter of the part. As another example, recall that the negative binomial random variable was represented as the sum of several geometric random variables. In this section, we develop results for random variables that are linear combinations of random variables.

De nition Given random variables X1, X2, p , Xp and constants c1, c2, p , cp, Y c1 X1 c2 X2 p cp Xp (5-36)

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Manipulating whole vectors and matrices at a time is not only concise and fast, but applicable to 3D graphics and some image processing algorithms.

Now, E(Y) can be found from the joint probability distribution of X1, X2, p , Xp as follows. Assume X1, X2, p , Xp are continuous random variables. An analogous calculation can be used for discrete random variables. E1Y2 1c1x1 c2 x2 p cp xp 2 fX1 X2 p Xp 1x1, x2, p , xp 2 dx1 dx2 p dxp

of the TargetAlert and the node with the maximum reading (i.e., which is closest to the target in this sense) should elect itself the leader. If the target can be detected at a maximum distance of d from the acoustic sensor on a particular sensor node, the maximum distance between nodes that can detect the same target is 2d. Hence, the output channel annotation for the sampling and thresholding task should require the target alert to be sent to all nodes within distance 2d of the node where the alert is produced. In this example, the 10m label implies that the sensing radius of the acoustic sensor is 5 meters. Now that the TargetAlert is disseminated from each node that has detected the object to every other node that has detected the object, it is necessary to define the mechanism of leader election in the ATaG program. In this case, we define a Leader task that is instantiated on each node of the network. This task consumes all TargetAlerts that are sent to the node from its neighboring nodes and also from the sampling task on the same node. The task maintains local state that consists of the readings received from the neighborhood. Using this information, the task can compare its own reading to the neighbor s readings; and if its own reading is the maximum, it declares itself the leader and also generates a TargetInfo notification to the Supervisor task on the root node. Now, communicationin the sensor network is inherently asynchronous because the sampling of the sensor on the nodes within range of the target is unlikely to be precisely synchronized. Hence, each node will generate its TargetAlert at different points in time. Also, multi-hop packet transmissions could introduce other delays in transmitting the TargetAlert.

Suitable only for input images, Pixel Bender includes image types for images with various numbers of color channels.

x1 fX1 X2 p Xp 1x1, x2, p , xp 2 dx1 dx2 p dxp x2 fX1 X2 p Xp 1x1, x2, p , xp 2 dx1 dx2 p dxp xp fX1 X2 p Xp 1x1, x2, p , xp 2 dx1 dx2 p dxp

Pretty much the only thing you can do with an image is sample it, using the three sample functions listed toward the end of Table 38-2. This yields a pixel of the corresponding number of channels.

cp Xp, c2E 1X2 2 p cp E 1Xp 2 (5-37)

Unfortunately, Pixel Bender on Flash doesn t support arrays. Instead, you could try to encode a series of oat values in an n 1 one-channel image (image1). Regions are not yet available.

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