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in Figure 86, we write X N(3, 2) It can be shown, but with some dif culty, that the total area of any normal curve is 1 Since the function is always positive, areas under the curve represent probabilities These cannot be calculated easily either Statistical calculators, however, can calculate these areas Here are some examples from this normal curve (a) P(2 X 6) = 0743303 (b) P(4 X 8) = 0239547 P(X > 4 and X > 2) P(X > 4) 023975 (c) P(X > 4|X > 2) = = = P(X > 2) P(X > 2) 076025 = 0315357 Textbooks commonly include a table of the standard normal curve, that is, N(0, 1) Since computers and calculators compute areas under the standard normal curve, we do not need to include such a table in this book The use of this standard normal curve to calculate areas under any normal curve is based on this fact Fact If X N( , ) and if Z = (X )/ , then Z N(0, 1) This means that areas under any normal curve can be calculated using a single, standard, normal curve For example, using our example (a) above, P(2 X 6) = P 2 3 X 3 6 3 = = 2 2 2

t l : E n a b l e ( P ) - > t 2 : E Re s et ( ) - > t 2 : D i s a b l e ( P ) - > t l : E Set ( )

Individuals: Enter the amount from Form 1040, line 39 If you are a nonresident alien, enter the amount from Form 1040NR, line 37 Estates and trusts: Enter your taxable income without the deduction 17 for your exemption Caution: If you figured your tax using the special rates on capital gains, see page 15 of the instructions Divide line 16 by line 17 If line 16 is more than line 17, enter 1 Individuals: Enter the amount from Form 1040, line 42 If you are a nonresident alien, enter the amount from Form 1040NR, line 40 Estates and trusts: Enter the amount from Form 1041, Schedule G, line 1a, or the total of Form 990-T, lines 36 and 37 Caution: If you are completing line 19 for separate category g (lump-sum distributions), see page 15 of the instructions Multiply line 19 by line 18 (maximum amount of credit) Enter the smaller of line 13 or line 20 If this is the only Form 1116 you are filing, skip lines 22 through 30 and enter this amount on line 31 Otherwise, complete the appropriate line in Part IV (see page 16 of the instructions)

In this example, t1 enables the predicate P, but before it has a chance to set the event, t2 comes along and disables P The result is that we wake up waiting threads although P is no longer true These threads must take care to re-evaluate P after being awakened to avoid proceeding blindly But unless they use additional data synchronization, this is impossible A nice codification of this relationship between state transitions and data and control synchronization was invented in the 1 970s (see Further Reading, Hansen; Hoare, 1 974) and is called monitors Each monitor implicitly has a critical region and may have one or more condition vari ables associated with it, each representing some condition (like P evaluat ing to true) for which threads may wish to wait In this sense, a condition variable is just a fancy kind of event All waiting and signaling of a monitor's condition variables must occur within the critical region of the monitor itself, ensuring data race protection When a thread decides to wait on a condition variable, it implicitly releases

} else { I I Ot herwi s e , we ' re f u l l j expand t h e q u e u e by II d o u b l i n g its s i z e ( ignoring ove rflow ) T [ ] newArray new T [ m_a rray Length l ] j for ( i nt i 0 j i < m_a rray Lengt h j i++) newArray [ i ] m_a rray [ ( i + head ) & m_ma s k ] j

= P ( 0707 11 Z 2 121 3) = 0743303 as before The fact that any normal curve can be transformed into a standard normal curve is a remarkable fact (and a fact not true for the other probability density functions we will encounter) There are, however, many other uses of the standard normal curve and we will meet these when we study other statistical topics

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