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(c) What is the probability that 4 of the 6 numbers chosen by a player appear in the state s sample (d) If a player enters one lottery each week, what is the expected number of weeks until a player matches all 6 numbers in the state s sample 3-95. Continuation of Exercises 3-86 and 3-87. (a) Calculate the nite population corrections for Exercises 3-86 and 3-87. For which exercise should the binomial approximation to the distribution of X be better (b) For Exercise 3-86, calculate P1X 12 and P1X 42 assuming that X has a binomial distribution and compare these results to results derived from the hypergeometric distribution. (c) For Exercise 3-87, calculate P1X 12 and P1X 42 assuming that X has a binomial distribution and compare these results to the results derived from the hypergeometric distribution. 3-96. Use the binomial approximation to the hypergeometric distribution to approximate the probabilities in Exercise 3-92. What is the nite population correction in this exercise

It s almost always a good idea to keep your classes open for extension. If, after you wrap up and deliver your toasting program, you nd that you need more toasting power, you might decide to create a toaster class with multiple resistors. A quick way to do this is to extend the HeatedResistorToaster class and add a second resistor, modifying the startCurrent() and stopCurrent() methods to support two resistors. To do this, you need to go back and make those methods protected instead of private. Whether private or protected, those methods are not available to

EXAMPLE 3-30

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Consider the transmission of n bits over a digital communication channel. Let the random variable X equal the number of bits in error. When the probability that a bit is in error is constant and the transmissions are independent, X has a binomial distribution. Let p denote the probability that a bit is in error. Let and pn. Then, E1x2 pn P1X x2 n a b P x 11 x p2 n

classes outside the Toaster class hierarchy, so you haven t lost anything, but you needed to modify a class that you should have just been able to extend. Unless you have a reason to hide a property or method from every class in the class hierarchy or you re sure the class will never be extended, you should use protected over private. Using protected variables makes code easy to extend, even if you don t foresee a reason to right now. Use protected to encourage extension and final to prohibit it.

Now, suppose that the number of bits transmitted increases and the probability of an error decreases exactly enough that pn remains equal to a constant. That is, n increases and p decreases accordingly, such that E(X) remains constant. Then, with some work, it can be shown that limnS P1X x2 e x!

The idea that you should be able to add on to your program without changing anything that s already written is a principle of object oriented design called the Open-Closed Principle. According to the principle, code you write should be open for extension but closed for modi cation. This recognizes that code that s done means code that s been thought about, written, and tested. A lot of work has gone into it. It also recognizes that you should be able to add functionality to programs after you consider them done, because even if you hate doing it, you always need to. Composition and inheritance are ways to add more functionality without touching code that already exists; you should favor this to modifying completed, tested code.

If a new global behavior is to be added to an existing program, the modifications to the existing code are essentially ad hoc The strong coupling of applicationlevel logic and system-level services such as resource management, routing, localization, etc, also results in high coding complexity Macroprogramming broadly refers to programming methodologies for sensor networks that allow the direct specification of aggregate behaviors The.

Also, because the number of bits transmitted tends to in nity, the number of errors can equal any nonnegative integer. Therefore, the range of X is the integers from zero to in nity. The distribution obtained as the limit in the above example is more useful than the derivation above implies. The following example illustrates the broader applicability.

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