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3.64. (a) Find the moment generating function of a random variable X having density function f (x) e x>2 0 0 x 2 otherwise

(b) Use the generating function of (a) to find the first four moments about the origin. 3.65. Find the first four moments about the mean in (a) Problem 3.43, (b) Problem 3.44. 3.66. (a) Find the moment generating function of a random variable having density function f (x) e e 0

Ans: 0:585 lagging An induction motor load of 2000 kVA has power factor 080 lagging Synchronous motors totaling 500 kVA are added and operated at a leading power factor If the overall power factor is then 090 lagging, what is the power factor of the synchronous motors Ans: 0:92 leading Find maximum energy (E) stored in the inductor of Example 1017(a) and show that it is greater than the sum of maximum stored energies when each source is applied alone E1 and E2 Ans: E 1:6 mJ, E1 306 mJ, E2 506 mJ The terminal voltage and current of a two-terminal circuit are V rms 120 V and I rms 30 608 A at f 60 Hz Compute the complex power Find the impedance of the circuit and its equivalent circuit made of two series elements.

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and (b) determine the first four moments about the origin. 3.67. In Problem 3.66 find the first four moments about the mean. e 1>(b 0 a) a x b . Find the kth moment about (a) the origin, otherwise

Ans: S 1800 j3117:7 VA, Z 2 j3:464 R jL!, R 2.

3.69. If M(t) is the moment generating function of the random variable X, prove that the 3rd and 4th moments about the mean are given by m3 m4 M-(0) M(iv)(0) 3Ms(0)Mr(0) 4M-(0)Mr(0) 2[Mr(0)]3 6Ms(0)[Mr(0)]2 3[Mr(0)]4

The XYZ Corporation is developing software for Tom and Sue s Bed and Breakfast Inn. This software is intended to maintain reservation information and help in monitoring expenses and pro ts. This SQA plan is for version 1.0 to be delivered to Tom and Sue by November 1. This SQA plan covers the development life cycle of the complete system from requirements speci cation through software testing.

3.71. Find the characteristic function of a random variable X that has density function f (x) e 1>2a 0 a u xu otherwise

, L 9:2 mH In the circuit of Fig. 10-29 the voltage source has e ective value 10 V at ! 1 rad/s and the current source is zero. (a) Find the average and reactive powers delivered by the voltage source. (b) Find the e ective value of the current in the resistor and the average power absorbed by it and the reactive powers in L and C. Show the balance sheet for the average and reactive powers between the source and R, L, and C. p Ans: a P 80 W, Q 60 var, (b) IR 5 2 A, PR 80 W, QC 160 var, QL 100 var, PR P and QL QC Q

3.72. Find the characteristic function of a random variable with density function f (x) e e x>2 0 0 x 2 otherwise 1, 2, . . . , n). Prove that the characteristic

CHAP. 10]

1 prob. 1>2 be independent random variables (k 1 prob. 1>2 function of the random variable X1 is [cos (v> !n)]n. X2 c 2n

p p In the circuit of Fig. 10-29, va 10 2 cos t and ib 10 2 cos 2t. (a) Find the average power delivered by each source. (b) Find the current in the p resistor and the average power absorbed by it. p Ans: a Pa Pb 80 W; b iR 2 10 cos t 26:58 2 10 cos 2t 63:48 , PR 160 W

3.74. Prove that as n S ` the characteristic function of Problem 3.73 approaches e the characteristic function and use L Hospital s rule.)

3.75. Let X and Y be random variables having joint density function f (x, y) e x 0 y 0 x 1, 0 otherwise y 1

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