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Fig. 9-40 9.42 Find Z in the parallel circuit of Fig. 9-41, if V 50:0 Ans: 5:0 308 30:08 V and I 27:9 57:88 A.

Theorem 3-8 If MX (t) is the moment generating function of the random variable X and a and b (b 2 0) are constants, then the moment generating function of (X a) > b is M(X

Consider two dice. Consider a rolling a 7 as an undesirable event that would make you lose a pot of $60. Calculate the risk probability and the risk impact of rolling a 7. Calculate the risk exposure. The risk probability is 6 cases out of 36 combinations, or 1/6. The risk impact is $60. The risk exposure is 1/6 times $60, or $10.

a)>b(t)

Fig. 9-41 9.43 Obtain the conductance and susceptance corresponding to a voltage V 85:0 current I 41:2 141:08 A. Ans: 0:471 S; 0:117 S (capacitive) 2058 V and a resulting

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A practical coil contains resistance as well as inductance and can be represented by either a series or parallel circuit, as suggested in Fig. 9-42. Obtain Rp and Lp in terms of Rs and Ls . Ans: Rp Rs !Ls 2 R2 ; Lp Ls 2 s Rs ! Ls

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Theorem 3-9 If X and Y are independent random variables having moment generating functions MX (t) and MY (t), respectively, then MX

Y (t)

Fig. 9-42 9.45 In the network shown in Fig. 9-43 the 60-Hz current magnitudes are known to be: IT 29:9 A, I1 22:3 A, and I2 8:0 A. Obtain the circuit constants R and L. Ans: 5:8 ; 38:5 mH

MX (t) MY (t)

Fig. 9-43

A technique that can be used to visualize the risks of alternatives is to build a risk decision tree. The top-level branch splits based on the alternatives available. The next split is based on the probabilities of events happening. Each leaf node has the risk exposure for that event. The sum of the risk exposures for all leafs under the top-level split gives the total risk exposure for that choice.

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Generalizations of Theorem 3-9 to more than two independent random variables are easily made. In words, the moment generating function of a sum of independent random variables is equal to the product of their moment generating functions. Theorem 3-10 (Uniqueness Theorem) Suppose that X and Y are random variables having moment generating functions MX (t) and MY (t), respectively. Then X and Y have the same probability distribution if and only if MX (t) MY (t) identically.

Fig. 9-44

If we let t iv, where i is the imaginary unit, in the moment generating function we obtain an important function called the characteristic function. We denote this by fX (v) It follows that fX(v)

Obtain the magnitude of the voltage VAB in the two-branch parallel network of Fig. 9-44, if XL is (a) 5 , (b) 15 , c 0 . Ans: 50 V, whatever XL

MX (iv)

E(eivX)

Fig. 9-45

A friend offers to play one of two betting games with you. Game A is that you toss a coin twice. He pays you $10 if you get two heads. You pay him $2 for each tail you toss. Game B is that you also toss a coin twice, but it costs you $2 to play and he pays you $10 if you get two heads. Which game should you play The risk decision tree is shown in Fig. 6-1. Both games total to $0.50. Thus, each time you play, your average gain is 50 cents. No matter which game you choose.

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Fig. 9-46

a eivx f(x)

(discrete variable)

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