3. Buses arrive to a bus stop according to an exponential distribution with rate

?= 4 busses/hour. If you arrived at 8:00 am to the bus stop,

a) what is the expected time of the next bus?

b) Assume you asked one of the people waiting for the bus about the arrival

time of the last bus and he told you that the last bus left at 7:40 am. What

is the expected time of the next bus?

4. Break downs occur on an old car with rate ?= 5 break-downs/month. The owner

of the car is planning to have a trip on his car for 4 days.

a) What is the probability that he will return home safely on his car.

b) If the car broke down the second day of the trip and the car was fixed, what is

the probability that he doesn't return home safely on his car.

5. Suppose that the amount of time one spends in a bank is exponentially distributed with

mean 10 minutes. What is the probability that a customer will spend more than 15

minutes in the bank? What is the probability that a customer will spend more than 15

minutes in the bank given that he is still in the bank after 10 minutes?

6. Suppose the lifespan in hundreds of hours, T, of a light bulb of a home lamp is

exponentially distributed with lambda = 0.2. compute the probability that the light bulb

will last more than 700 hours Also, the probability that the light bulb will last more than

900 hours

7. Let X = amount of time (in minutes) a postal clerk spends with his/her customer. The

time is known to have an exponential distribution with the average amount of time equal

to 4 minutes.

a) Find the probability that a clerk spends four to five minutes with a randomly selected

customer.

b) Half of all customers are finished within how long? (Find median)

c) Which is larger, the mean or the median?

8. On the average, a certain computer part lasts 10 years. The length of time the computer

part lasts is exponentially distributed.

a) What is the probability that a computer part lasts more than 7 years?

b) On the average, how long would 5 computer parts last if they are used one after

another?

c) Eighty percent of computer parts last at most how long?

d) What is the probability that a computer part lasts between 9 and 11 years?

9. Suppose that the length of a phone call, in minutes, is an exponential random variable

with decay parameter = 1/12 . If another person arrives at a public telephone just before

you, find the probability that you will have to wait more than 5 minutes. Let X = the

length of a phone call, in minutes. What is median mean and standard deviation of X?

Another IFR turbine is to be built to develop 250 kW of shaft power from a gas flow of 1.1 kg/s.

The inlet stagnation temperature, T01, is 1050 K, the number of rotor blades is 13, and the outlet

static pressure, p3, is 102 kPa. At rotor exit the area ratio, ? r3h/r3s 0.4, and the velocity

ratio, cm3/U2 0.25. The shroud to rotor inlet radius, r3s/r2, is 0.4. Using the optimum efficiency

design method, determine

(i) the power ratio, S, and the relative and absolute flow angles at rotor inlet;

(ii) the rotor blade tip speed;

(iii) the static temperature at rotor exit;

(iv) the rotor speed of rotation and rotor diameter.

Evaluate the specific speed, ?s. How does this value compare with the optimum value of specific

speed determined in Figure 8.15?

14. Using the same input design data for the IFR turbine given in Problem 5 and given that the totalto-static efficiency is 0.8, determine

(i) the stagnation pressure of the gas at inlet;

(ii) the total-to-total efficiency of the turbine.

15. An IFR turbine is required with a power output of 300 kW driven by a supply of gas at a stagnation pressure of 222 kPa, at a stagnation temperature of 1100 K, and at a flow rate of 1.5 kg/s.

The turbine selected by the engineer has 13 vanes and preliminary tests indicate it should have a

total-to-static efficiency of 0.86. Based upon the optimum efficiency design method sketch the

appropriate velocity diagrams for the turbine and determine

(i) the absolute and relative flow angles at rotor inlet;

(ii) the overall pressure ratio;

(iii) the rotor tip speed.

16. For the IFR turbine of the previous problem the following additional information is made available:

cm3=U2 0:25,w3=w2 2:0,r3s=r2 0:7 and ? 0:4:

Again, based upon the optimum efficiency design criterion, determine,

(i) the rotor diameter and speed of rotation;

(ii) the enthalpy loss coefficients of the rotor and the nozzles given that the nozzle loss coefficient is (estimated) to be one quarter of the rotor loss coefficient

For problems 18 - 25, assume X and Y are independent random variables with the following distributions: X -1 0 1 2 P(X) 0.3 0.1 0.5 0.1 Y 2 13 5 P(Y) 0.6 0.3 0.1 18. Find the mean, variance, and standard deviation of X. 19. Find the mean, variance, and standard deviation of Y. 20. Let W= 3 +2 X. Find the mean, variance, and standard deviation of W. 21. Let W = X+ Y. Find the mean, variance, and standard deviation of W.4) In healthy adult males the sciatic nerve conduction velocity values are normally distributed with a mean of 65 cm/msec and a standard deviation of 5 cm/msec. The conduction velocities of 16 subjects admitted to the poison control center of a metropolitan hospital with a diagnosis of methylmercury poisoning had a mean conduction velocity of 55 cm/msec and a variance of 49 (cm/msec)2. (a) Do these data provide sufficient evidence to indicate that the conduction velocities are significantly slower in the poisoned individuals? (b) Do these data provide sufficient evidence to indicate that poisoned individuals are more variable in their sciatic nerve conduction velocities than normal individuals?Mean, Variance, and Standard Deviation for the Binomial Distribution The mean, variance, and the standard deviation of a variable that has the binomial distribution can be found by using the following formulas. Mean : H = n . p Variance: 02 =n. p.q Standard deviation: o = n . p.q Example # 23: Find the mean, variance, and standard deviation for the number of heads when ten coins are tossed. Example # 24: If 3% of calculators are defective, find the mean, variance, and standard deviation of a lot of 300 calculators.5.54 Suppose the random variable X has the probability distribution given in the table below. 2 3 5 7 11 13 p(x) | 0.15 0.25 0.15 0.10 0.30 0.05 a. Find the mean, variance, and standard deviation of X. b. Suppose the random variable Y is defined by Y = 2X + 1. Find the mean, variance, and standard deviation of Y. c. Suppose the random variable W is defined by W = X' + 1. Find the mean, variance, and standard deviation of W. 5 55 Sunn