23-2. An intercom system master station provides music to six hospital rooms. Th

Question

23-2. An intercom system master station provides music to six hospital rooms. The probability that any one room will be switched on and draw power at any time is 0.4. When on, a room draws 0.5 W. (a) Find and plot the density and distribution functions for the random vari able "power delivered by the master statio (b) If the master-station amplifier is overloaded when more than 2 W is demanded, what is its probability of overload? 3.1 1. A discrete random variable X has possible values xi- 1, 2, 3, 4, 5, which occur with probabilities 0.4, 0.25, 0.15, 0.1, and 0.1, respectively. Find the mean value X = E of X, 3.1-2. The natural numbers are the possible values of a random variable X: that is, x,-n, n= 1,2, These numbers occur with probabilities P(%)=( Find the expected value of X. 3.1-4. (a) Find the average amount the gambler in Problem 2.1-13 can expect to win (b) What is his probability of winning on any given playing of the game?

Explanation / Answer

3.1-1)

E(X) = Sum(X*P(X)) = 0.4 + 0.5 + 0.45 + 0.4 + 0.5 = 2.25

3.1-2)

E(X) = Sum(X*P(X))

= 1*(1/2) + 2* (1/2)2 + 3*(1/2)3 + 4*(1/2)4+.. upto infinity => 1)

1/2*E(X) = 1/2* (1*(1/2) + 2* (1/2)2 + 3*(1/2)3 + 4*(1/2)4+.. )

Thus,

1/2*E(X) = 1*(1/2)2 + 2* (1/2)3 + 3*(1/2)4 + 4*(1/2)5+.. => 2)

substract 2) from 1)

we get

E(X) - 1/2*E(X) = (1/2)+ (1/2)2 + (1/2)3 + (1/2)4+..

Thus, 1/2*E(X) = (1/2)+ (1/2)2 + (1/2)3 + (1/2)4+..

RHS is geometric progression with r=1/2

Sum is given as a*1/(1-r) .. {a= 1st term = 1/2, r= common ratio = 1/2}

= (1/2)/(1-(1/2)) = 1

Thus, 1/2*E(X) = 1

E(X) = 2

X 1 2 3 4 5 P(X) 0.4 0.25 0.15 0.1 0.1 X*P(X) 0.4 0.5 0.45 0.4 0.5

Related Questions

Navigate

• Browse (All)
• Browse 2
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.