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Similar remarks can be made for two discrete random variables. In such cases (43) and (46) are replaced by mX a a xf(x, y)

Consider the risks identified in the risk identification problem. Suggest an approach for either decreasing the probability or decreasing the impact or both. Risk Hardware not available Requirements incomplete Decrease Probability Accelerate dev. of hardware Increase review of requirements Increase sta training, hire experts Design for reliability Decrease Impact Build simulator

Fig. 9-53

a a yf(x, y)

(47) (48)

Obtain the Theveinin and Norton equivalent circuits at terminals ab for the network of Fig. 9-53. Ans: V 0 11:18 93:438 V; I 0 2:24 56:568 A; Z 0 5:0 36:878 In the circuit of Fig. 9-54, v1 10 V and v2 5 sin 2000t. Ans: i 1 0:35 sin 2000t 458 Find i.

mY) f(x, y)

Fig. 9-54

where the sums are taken over all the discrete values of X and Y. The following are some important theorems on covariance. Theorem 3-14 Theorem 3-15 sXY E(XY ) E(X)E(Y ) E(XY ) mXmY (49)

If X and Y are independent random variables, then sXY Cov (X, Y ) Var (Y ) s2 Y 0 2 Cov (X, Y ) 2sXY (50) (51) (52) (53)

Fig. 9-55

Use of specialized methodologies Problems achieving required reliability Retention of key people Underestimating required e ort The single potential customer goes bankrupt

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Theorem 3-16 or Theorem 3-17

In the circuit of Fig. 9-55, v1 6 cos !t and v2 cos !t 608). Find vA if ! 2 rad/sec. KCL at node A in the phasor domain. Ans: vA 1:11 sin 2t

The converse of Theorem 3-15 is not necessarily true. If X and Y are independent, Theorem 3-16 reduces to Theorem 3-7.

If X and Y are independent, then Cov(X, Y) sXY for example, when X Y, then Cov(X, Y) sXY of the variables X and Y given by r 0. On the other hand, if X and Y are completely dependent, sX sY. From this we are led to a measure of the dependence sXY sX sY

In the circuit of Problem 9.59 nd phasor currents I1 and I2 drawn from the two sources. phasor KVL to the loops on the left and right sides of the circuit. Ans: I1 508 100:48; I2 1057 1458, both in mA

(54)

Ans:

We call r the correlation coefficient, or coefficient of correlation. From Theorem 3-17 we see that 1 r 1. In the case where r 0 (i.e., the covariance is zero), we call the variables X and Y uncorrelated. In such cases, however, the variables may or may not be independent. Further discussion of correlation cases will be given in 8.

A risk management plan must include an identi er, a description of the risk, an estimate of risk probability, an estimate of risk impact, a list of mitigation strategies, contingency plans, risk triggers (to determine when contingency plans should be activated), and responsible individuals. Additional elds might include current and/or past status of related metrics.

(2.8)2 50

In the circuit of Fig. 9-55, v1 V1 cos 0:5t 1 and v2 V2 cos 0:5t 2 . Find the current through the 4 H inductor. Ans: i V2 =4 sin 0:5t 2 V1 =3 sin 0:5t 1

where we have used the sample standard deviations as estimates of s1 and s2. Then Z # # X1 X2 sX1 X2 68.2 67.5 0.53 1.32

In the circuit of Fig. 9-55, v1 V1 cos t 1 and v2 V2 cos t 2 . Ans: vA 1, unless V1 V2 0, in which case vA 0

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