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A sinusoidal voltage v Vm cos !t, applied across an impedance Z jZj , establishes a current i Im cos !t . The power delivered to the impedance at time t is p t vi Vm Im cos !t cos !t 1 Vm Im cos cos 2!t 2 Veff Ieff cos cos 2!t Veff Ieff cos Veff Ieff cos 2!t 2 p p where Veff Vm = 2, Ieff Im = 2, and Ieff Veff =jZj. The instantaneous power in (2) consists of a sinusoidal component Veff Ieff cos 2!t plus a constant value Veff Ieff cos which becomes the average power Pavg . This is illustrated in Fig. 10-4. During a portion of one cycle, the instantaneous power is positive which indicates that the power ows into the load. During the rest of the cycle, the instan-

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1. Why is risk management important 2. Consider driving to the airport to catch a plane on an airline that you have not used before. What risks might be unique to this trip to the airport, and which ones might be managed as part of the normal trip to the airport

EXAMPLE 3.5 The average travel time to a distant city is c hours by car or b hours by bus. A woman cannot decide whether to drive or take the bus, so she tosses a coin. What is her expected travel time Here we are dealing with the joint distribution of the outcome of the toss, X, and the travel time, Y, where Y Ycar if X 0 and Y Ybus if X 1. Presumably, both Ycar and Ybus are independent of X, so that by Property 1 above

CHAP. 10]

E(Ycar u X l)

Fig. 10-3

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E(Ycar) 1)

E(Ybus)

taneous power may be negative, which indicates that the power ows out of the load. of power during one cycle is, however, nonnegative and is called the average power.

Then Property 2 (with the integral replaced by a sum) gives, for a fair coin, E(Y) E(Y u X 0)P(X 0) E(Y u X 1)P(X 1) c 2 b

1. Analyze the potential risks for the dental o ce problem in 4, Problem 2. Classify the risks as normal or unique to this project. 2. Consider a project that has 0.5 percent probability of an undetected fault that would cost the company $100,000 in nes. Calculate the risk exposure.

EXAMPLE 10.3 A voltage v 140 cos !t is connected across an impedance Z 5 608. The voltage v results in a current i 28 cos !t 608 . Then, p t vi 140 28 cos !t cos !t 608 980 1960 cos 2!t 608

In a similar manner we can define the conditional variance of Y given X as E[(Y where m2 E(Y u X m2)2 u X

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The instantaneous power has a constant component of 980 W and a sinusoidal component with twice the frequency of the source. The plot of p vs. t is similar to that in Fig. 10-4 with =3.

x). Also we can define the rth conditional moment of Y about any value a given X as E[(Y a)r u X x] 3 `(y a)r f (y u x) dy (57)

The net or average power Pavg hp t i entering a load during one period is called the real power. Since the average of cos 2!t over one period is zero, from (2) we get Pavg Veff Ieff cos 3

An important theorem in probability and statistics that reveals a general property of discrete or continuous random variables having finite mean and variance is known under the name of Chebyshev s inequality. Theorem 3-18 (Chebyshev s Inequality) Suppose that X is a random variable (discrete or continuous) having mean m and variance s2, which are finite. Then if P is any positive number, P(uX or, with P ks, P(uX mu ks) 1 k2 (59) mu P) s2 P2 (58)

2 in Chebyshev s inequality (59), we see that P (u X mu 2s) 0.25 or P( u X mu 2s) 0.75

Fig. 10-4

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