Question:

Over a very long period of time, it has been noted that on Friday's 25% of the customers

at the drive-in window at the bank make deposits. What is the probability that it takes 4

customers at the drive-in window before the first one makes a deposit.

2. It is estimated that 45% of people in Fast-Food restaurants order a diet drink with their

lunch. Find the probability that the fourth person orders a diet drink. Also find the

probability that the first diet drinker of th e day occurs before the 5th person.

3. What is the probability of rolling a sum of seven in fewer than three rolls of a pair of

dice? Hint (The random variable, X, is the number of rolls before a sum of 7.)

4. In New York City at rush hour, the chance that a taxicab passes someone and is

available is 15%. a) How many cabs can you expect to pass you for you to find one that

is free and b) what is the probability that more than 10 cabs pass you before you find

one that is free.

5. An urn contains N white and M black balls. Balls are randomly selected, one at a time,

until a black ball is obtained. If we assume that each selected ball is replaced before the

next one is drawn, what is;

a) the probability that exactly n draws are needed?

b) the probability that at least k draws are needed?

c) the expected value and Variance of the number of balls drawn?

6. In a gambling game a player tosses a coin until a head appears. He then receives $2n ,

where n is the number of tosses.

a) What is the probability that the player receives $8.00 in one play of the game?

b) If the player must pay $5.00 to play, what is the win/loss per game?

7. An oil prospector will drill a succession of holes in a given area to find a productive

well. The probability of success is 0.2.

a) What is the probability that the 3rd hole drilled is the first to yield a productive well?

b) If the prospector can afford to drill at most 10 well, what is the probability that he will

fail to find a productive well?

8. A well-travelled highway has itstraffic lights green for 82% of the time. If a person

travelling the road goes through 8 traffic intersections, complete the chart to find a) the

probability that the first red light occur on the nth traffic light and b) the cumulative

probability that the person will hit the red light on or before the nth traffic light.

9. An oil prospector will drill a succession of holes in a given area to find a productive

well. The probability of success is 0.2.

a) What is the probability that the 3rd hole drilled is the first to yield a productive well?

b) If the prospector can afford to drill at most 10 well, what is the probability that he will

fail to find a productive well?

1. Calculate the Poisson distribution whose ? (Average Rate of Success)) is 3 & X (Poisson

Random Variable) is 6.

2. Customers arrive at a checkout counter according to a Poisson distribution at an average

of 7 per hour. During a given hour, what are the probabilities that

a) No more than 3 customers arrive?

b) At least 2 customers arrive?

c) Exactly 5 customers arrive?

3. Manufacturer of television set knows that on an average 5% of their product is defective.

They sells television sets in consignment of 100 and guarantees that not more than 2 set

will be defective. What is the probability that the TV set will fail to meet the guaranteed

quality?

4. It is known from the past experience that in a certain plant there are on the average of 4

industrial accidents per month. Find the probability that in a given year will be less that 3

accidents.

5. Suppose that the change of an individual coal miner being killed in a mining accident

during a year is 1.1499. Use the Poisson distribution to calculate the probability that in

the mine employing 350 miners- there will be at least one accident in a year.

6. The number of road construction projects that take place at any one time in a certain city

follows a Poisson distribution with a mean of 3. Find the probability that exactly five road

construction projects are currently taking place in this city. (0.100819)

7. The number of road construction projects that take place at any one time in a certain city

follows a Poisson distribution with a mean of 7. Find the probability that more than four

road construction projects are currently taking place in the city. (0.827008)

Exercise 7. [10 points] Estimators' Properties You know a random variable X has IE[X] = ,u. The value for its variance V[X] = 02 is nite but unknown and you want to estimate it. a) Consider the following estimator: 1 n 62 := 2X? n i=1 a) Use the weak law of large numbers to construct the probability limit of 62. That is 62 ? Problem 3 (40 points) Let {X, } be any sequence of random variables. Let Yr = _ EF, X,. A necessary and sufficient condition for the sequence {X, } to satisfy the weak law of large numbers is that E n -+ 0 as n -+ 0o 1 + Y2 First show that for any two positive numbers a, b, a 2 b > 0 a 1+b 1 ta b 21Problem 5. Let X1, X2, . .. be an infinite sequence of Lid. random variables. Let f be a function. Prove the weak law of large numbers for {f(X,.)}. Specifically, show that if Sa = f(XI ) + f (X2 ) + ... + / (X.). then for any 0. P Sn You need to prove this result from scratch. You cannot use the law of large numbers for the sequence {X.). You can2. Use Chebyshev's inequality to prove the weak law of large numbers, which goes as follows. Let X1, X2, . .. be i.i.d. with mean / and Variance o'. Then for any & > 0, Xi+ X2+ . . . Xn lim P -H >E =0 n-+00 n Remark. This is related to a third notion of convergence called convergence in prob- ability. Look it up in Wikipedia for the basic definitions