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3. Consider the use of an additional review for Problem 2 that would cost $100 but eliminate such a fault 50 percent of the time. Calculate this new risk exposure with using the additional review. Is the additional review approach better 4. What would change in Problem 3 if the additional review was only e ective 10 percent of the time 5. Build a decision tree for the problem in Example 6.3 if in Game A, the payo was $5.00 and if the cost of playing in Game B was $4.00. Should you play either game 6. Company X has historical data that shows a normal error rate of 0.0036 errors per KLOC. A study of a new review technique shows that it costs $1000 per 100 KLOC and decreases the number of errors by 50 percent. Assume that each error costs the company an average of $10,000. The current project is estimated to be 50 KLOC in size. Calculate risk exposure for each approach. Is the new review technique worth doing

In words, the probability of X differing from its mean by more than 2 standard deviations is less than or equal to 0.25; equivalently, the probability that X will lie within 2 standard deviations of its mean is greater than or equal to 0.75. This is quite remarkable in view of the fact that we have not even specified the probability distribution of X.

If Z R jX jZj , then cos R=jZj and Pavg may be expressed by Pavg Veff Ieff or or Pavg R jZj 4

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The following theorem, called the law of large numbers, is an interesting consequence of Chebyshev s inequality. Theorem 3-19 (Law of Large Numbers): Let X1, X2, . . . , Xn be mutually independent random variables (discrete or continuous), each having finite mean m and variance s2. Then if Sn X1 X2 c ), Xn(n 1, 2, c Sn lim P 2 n m2 P 0 (60)

Since Sn > n is the arithmetic mean of X1, . . . , Xn, this theorem states that the probability of the arithmetic mean Sn > n differing from its expected value m by more than P approaches zero as n S ` . A stronger result, which we might expect to be true, is that lim Sn >n m, but this is actually false. However, we can prove that nS` lim Sn >n m with probability one. This result is often called the strong law of large numbers, and, by contrast, nS` that of Theorem 3-19 is called the weak law of large numbers. When the law of large numbers is referred to without qualification, the weak law is implied.

(5) (6)

As we have already seen, the mean, or expectation, of a random variable X provides a measure of central tendency for the values of a distribution. Although the mean is used most, two other measures of central tendency are also employed. These are the mode and the median. 1. MODE. The mode of a discrete random variable is that value which occurs most often or, in other words, has the greatest probability of occurring. Sometimes we have two, three, or more values that have relatively large probabilities of occurrence. In such cases, we say that the distribution is bimodal, trimodal, or multimodal, respectively. The mode of a continuous random variable X is the value (or values) of X where the probability density function has a relative maximum.

The average power is nonnegative. It depends on V, I, and the phase angle between them. When Veff and Ieff are given, P is maximum for 0. This occurs when the load is purely resistive. For a purely reactive load, j j 908 and Pavg 0. The ratio of Pavg to Veff Ieff is called the power factor pf. From (3), the ratio is equal to cos and so pf Pavg Veff Ieff 0 pf 1 7

1 1 2. MEDIAN. The median is that value x for which P(X x) 2 and P(X x) 2. In the case of a con1 tinuous distribution we have P(X x) 2 P(X x), and the median separates the density curve into two parts having equal areas of 1 > 2 each. In the case of a discrete distribution a unique median may not exist (see Problem 3.34).

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