STA104/MTH135: Probability

Homework #9: Review for Second Midterm

Due: Wednesday, Nov 14

Review Problems

1. Pick two points at random, uniformly from the unit square. Find the expected area of the smallest circle that contains both points.
2. Pick a point at random from the triangle with corners (0,0), (1,0), and (0,2). Find:
   • The joint density function for X and Y;
   • The conditional expectation of X, given Y=1;
   • The expected value of XY;
   • The probability P[X<Y];
   • The density function for Z=Y/X.
3. How many times would you have to roll three fair dice, on average, before finding the same number on all three?
4. In each example below, give a probability distribution (including the values of any parameters) for the quantity indicated; justify your choices:
   • The number of patients who respond favorably to a new drug, in a trial with 25 subjects, if about 20% of the population responds;
   • The number of typographical errors on a manuscript page, if pages have about 1000 characters and about 0.1% of keystrokes are incorrect;
   • The number of pages with NO typographical errors in the entire 100-page document;
   • The page number of the first page that does have a typographical error, if pages are numbered starting at 1;
   • The length of time you wait for the East-West campus bus, if about twelve buses come per hour on average;
   • The number of buses that arrive during the 10 minutes you wait for your friend;
   • The number your watch's sweep-second-hand is pointing to, when you look at it in hoping class is nearly over;

Good Luck!