Statistical Methods

Homework 2:

Note 1: Homeworks are due at the beginning of the "due date" class. Late hw will not be accepted, except in extraordinary circumstances.

Note 2: If you can't solve some of the problems, please come to the office hours, or email us a very specific and short question. Email will not work well or at all for involved or unclear questions.

For this homework, please complete the following problems:

• Problem 1
  

  One out of 500 people has disease D. You're a doctor and suspect that one of your patients may have that disease. You order a diagnostic test T for that patient. The test is not perfect - it accurately claims the disease is "present" in 80% of the patients who actually have it, and accurately declares the disease as "absent" in 70% of the patients who indeed don't have the disease.

  • a) If the test result comes back negative (the test says "absence of disease"), what does that mean for your patient -- how likely is he to actually have disease D?
    
  • b) How about if the test comes back positive (the test says "disease") -- how likely is he to actually have disease D then?
    
  • c) If the test comes back positive (the test says "disease"), is your patient more likely to actually have disease D than not?
    
  • d) If the test comes back negative (the test says "absence of disease"), is your patient more likely to not have disease D than have it?
    
    

• Problem 2
  

  This problem is about probabilities of events, where events are treated as sets. Thus, if the probability of event J is p, then the probability of the complement of J (ie, probability of everything other than the event J) is (1-p). We denote the complement of J with the "prime" symbol, as J', and refer to it as J-prime.
  

  • a) If events J and K are independent, what can we say about J' and K -- are they independent too?
    
  • b) If events J and K are independent, what can we say about J' and K' -- are they independent too?
    
  • c) If events A, B and C are mutually disjoint, what can we say about A', B', and C' -- are they mutually disjoint too?
    
  • d) If events A, B and C are mutually disjoint, can A and C be independent?
    
  • e) If events A, B and C are mutually disjoint, are A' and C' independent?
    
  • f) If E is independent of F, and F is independent of G, is E independent of G?
    
  
  

• Problem 3
  

  A certain lab machine has 12 rings. With each use, these rings can fail, and an oil leak occurs. The probability of any ring failing during any machine use is 2%. The rings are independent, and the failure of one ring does not impact the probability of failing for other rings. The machine gets serviced after each use, so any damaged rings are repaired after each use.
  
  

  If 6 or more rings fail during any single use, the entire machine will shut down.
  
  
  • a) What is the probability that the machine will shut down within the first 5 uses?
    
  • b) What is the probability that the machine will not shut down within the first 10 uses?
    
  • c) What is the probability that no rings will fail within the first 10 uses?
    
  
  

• Problem 4
  

  A traffic office wishes to monitor the number of vehicles crossing a certain bridge. They start by counting the vehicles crossing the bridge during any given day, a random variable which we will denote as X.
  

  • a) What family of distributions will they most likely have for X?
    
  • b) If an average of 30 cars cross the bridge per day, what specific member of the above family of distributions are they most likely working with?
    
  • c) Assuming the distribution from part (b), what is the probability that less than 10 cars will cross the bridge on any given day?
  • d) Assuming the distribution from part (b), what is the probability that no cars will cross the bridge on any given day?