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Week 1 Homework
Due Jan 23 at 11:59pm Points 12 Questions 12 Available Jan 16 at 8am - Jan 26 at 11:59pm Time Limit None
Instructions
This quiz was locked Jan 26 at 11:59pm.
Attempt History
Attempt Time Score LATEST Attempt 1 4,236 minutes 12 out of 12
Score for this quiz: 12 out of 12 Submitted Jan 23 at 10:57pm This attempt took 4,236 minutes. Correct answer
Question 1 1 / 1 pts
a. -11.625 sec b. 2 sec c. 5.375 sec
Set
Please answer all the questions below.
(Lesson 1.3: Deterministic Model.) Suppose you throw a rock off a cliff having height = 1000 feet. You're a strong bloke, so the initial downward velocity is = -100 feet/sec (slightly under 70 miles/hr). Further, in this neck of the woods, it turns out there is no friction in the atmosphere - amazing! Now you remember from your Baby Physics class that the height after time is
When does the rock hit the ground?
and solve for t. Quadratics are easy:
which we take as the answer since the negative answer doesn't make practical sense.
d. 11.625 sec e. 10 sec
Set
and solve for t. Quadratics are easy:
which we take as the answer since the negative answer doesn't make practical sense. Correct answer
Question 2 1 / 1 pts
a. 1 b. 2/ c. 0. d. 0.
At time , we have
(Lesson 1.3: Stochastic Model.) Consider a single-server queueing system where the times between customer arrivals are independent, identically distributed Exp(λ = 2/hr) random variables; and the service times are i.i.d. Exp(μ = 3/hr). Unfortunately, if a potential arriving customer sees that the server is occupied, he gets mad and leaves the system. Thus, the system can have either 0 or 1 customer in it at any time. This is what’s known as an M/M/1/1 queue. If denotes the probability that a customer is being served at time t, trust me that it can be shown that
If the system is empty at time 0, i.e., , what is the probability that there will be no people in the system at time 1 hr?
a. We put $5000 into a savings account paying 2% continuously compounded interest per year, and we are interested in determining the account's value in 5 years.
b. We are interested in investing one half of our portfolio in fixed-interest U.S. bonds and the remaining half in a stock market equity index. We have some information concerning the distribution of stock market returns, but we do not really know what will happen in the market with certainty.
c. We have a new strategy for baseball batting orders, and we would like to know if this strategy beats other commonly used batting orders (e.g., a fast guy bats first, a big, strong guy bats fourth, etc.). We have information on the performance of the various team members, but there’s a lot of randomness in baseball.
d. We have an assembly station in which “customers” (for instance, parts to be manufactured) arrive every 5 minutes exactly and are processed in precisely 4 minutes by a single server. We would like to know how many parts the server can produce in a hour.
e. Consider an assembly station in which parts arrive randomly, with independent exponential interarrival times. There is a single server who can process the parts in a random amount of time that is normally distributed. Moreover, the server takes random breaks every once in a while. We would like to know how big any line is likely to get.
f. Suppose we are interested in determining the number of doctors needed on Friday night at a local emergency room. We need to ensure that 90% of patients get treatment within one hour.
(a) and (d) do not require simulation, since we can easily “solve” those models with a simple equation or two. (b), (c), (e), and (f) will likely require simulation. Correct answer
Question 5 1 / 1 pts
a. 1/(49 · 50) b. 1/
b. Let’s call the two guys A and B. Whatever A’s birthday is, the probability that B matches it is 1/50.
(Lessons 1.6 and 1.7: Baby Examples.) The planet Glubnor has 50-day years.
Suppose there are 2 Glubnorians in the room. What’s the probability that they’ll have the same birthday?
Let’s try it another way. The total number of ways that two people can have birthdays is 50 × 50 =
- The total number of ways that they can have two different birthdays is 50 × 49 = 2450. Thus,
c. 1/ d. 2/
Let’s call the two guys A and B. Whatever A’s birthday is, the probability that B matches it is 1/50.
Let’s try it another way. The total number of ways that two people can have birthdays is 50 × 50 =
- The total number of ways that they can have two different birthdays is 50 × 49 = 2450. Thus,
Correct answer
Question 6 1 / 1 pts
a. 1/ b. 2/ c. 1/(49 · 50) d. 0.
d. I admit that this involves a teensy bit of probability (that you will eventually review in Module 2), but it should be easy enough. Mimicking the previous question, we have
d. I admit that this involves a teensy bit of probability (that you will eventually review in Module 2), but it should be easy enough. Mimicking the previous question, we have
Correct answer
Question 7 1 / 1 pts
(Lessons 1.6 and 1.7: Baby Examples.) The planet Glubnor has 50-day years.
Now suppose there are 3 Glubnorians in the room. (They’re big, so the room is getting crowded.) What’s the probability that at least two of them have the same birthday?
a. 18 b. 25 c. 33
Here is the sequence of relevant events
d. 45
(c). Here is the sequence of relevant events
Correct answer
Question 10 1 / 1 pts
a. 0 b. 1/ c. 7/
(Lesson 1.8: Generating Randomness.) Suppose we are using the (awful) pseudo-random number generator
with starting value ("seed"). Find the second PRN,
We have
d. 3
We have
and the answer is (c). Correct answer
Question 11 1 / 1 pts
a. 352515241 b. 16808 c. 1335380034 This is actually not quite so easy as it may seem, since you have to be a little careful not to lose significant digits. We'll learn more about this in Module 6. In any case,
where I multiplied the big numbers and took the mod with the help of Excel.
d. 12345679
This is actually not quite so easy as it may seem, since you have to be a little careful not to lose significant digits. We'll learn more about this in Module 6. In any case,
where I multiplied the big numbers and took the mod with the help of Excel.
(Lesson 1.8: Generating Randomness.) Suppose we are using the "decent" pseudo-random number generator
with seed = 12345678. Find the resulting integer. Feel free to use something like Excel if you need to.
