NOTE: This homework is due on Thursday, March 5. These problems are supplemental to Homework 5. The idea is to be fully caught up so we can enjoy Spring Break.


  1. Find the value for the following infinite sum. Round to 5 decimal places. Just put down the answer (you do not need to show your work). \[1+\frac{10}{1!}+ \frac{10^2}{2!}+\frac{10^3}{3!} + \frac{10^4}{4!} + \cdots\] The math formula for the above infinite sum is \[\sum_{i=0}^{\infty} \frac{10^i}{i!}\]



2. Robin occasionally goes biking on the American Tobacco trail. When you are passing a runner you are supposed to call out ‘’on your left’‘but Robin always forgets. Suppose he passes 200 runners this Sunday. Further suppose each person he passes has a chance of \(.01\) of yelling at him for not calling out ``on your left’’.

What is the chance that Robin is yelled at

  1. Exactly twice?
  2. More than (\(>\)) twice?

Use Poisson approximation for both parts.


3. By NC law, you are supposed to be using headlights when you are using windshield wipers owing to ``inclement weather’’. Shankar is driving to work on a rainy day and starts inspecting cars that pass him. Suppose each car he passes forgets to turn on the headlights with probability \(.05\) (independent across different cars). Further suppose he passes 50 cars on the way to work. Let \(X\) be the number of cars (out of 50) that forget to turn on their lights.

  1. X has a Binomial distribution. What are the parameters \(n\) and \(p\)?
  2. Using the Binomial pmf calculate \(P(X=2)\) namely Shankar passes exactly two cars whose headlights are not turned on. Round to 4 decimal places.
  3. Use Poisson approximation to calculate the same probability. Round to 4 decimal places.


4. The number of accidents on a certain section of I-40 averages \(4\) accidents per week day independent across week days. Assuming the number of accidents on a day follows a Poisson distribution

  1. What is the probability there are no car accidents on that stretch on Monday?
  2. What is the probability that out of the 5 weekdays next week, there are 3 days with exactly 4 accidents and 2 days where the number of accidents are different from 4 (could be less or more).

Hint: In the second part you will eventually need to use the Binomial distribution.


5.The industrial city of Isengard has major air pollution problems owing to the Mayor Saruman founding a large number of Orc-based industries. The air regulation authority of Isengard says an ``event of poor air quality’’ occurs if the Particulate matter per million exceeds 10 units. Assume that this can be modeled as a Poisson process with rate of \(\lambda =2\) per month.

What is the probability of at most 4 events of poor air quality in the next \(t=4.5\) months?


6. Suppose the university wifi has ``technical issues’’ according to a Poisson process with rate \(\lambda =4\) per day (so per 24 hours).

  1. What is the probability that there is at least one failure event in the next one hour.
  2. What is the probability that there is exactly one failure event between 8:00 am and 10:00 am tomorrow and exactly one failure event between 3:00 pm and 4:30 pm. (You may make the appropriate independence assumptions for a Poisson Process. We assume failure rate is the same at different times of day.)


7. Suppose \(X\) is a Poisson random variable with mean \(\lambda\). If \(P[X=1|X\leq 1] =.9\) what is the value of \(\lambda\)?


8. Write down the answers to the following integrals. You do not need to show work (but for the course you need to know how to do basic integrals so I strongly urge you to do them by hand. Don’t use Wolfram alpha till you know at least conceptually how to do the integrals below). Round to 4 decimal places.

  1. \(\int_0^4 x^3 dx\).
  2. \(\int_0^1 e^{x} dx\).
  3. For what value of \(c\) does one have \(c\int_0^1 (2-x^2) dx = 1\).