1. consider a sphere or radius r in d dimensions. Show the fraction of volume which lies between r and r + e where e is small is
1-(1-e/r)^d What is the limit of this quantity as d goes to infinity?
2. Fred is giving out samples of dog food. He makes calls door to door and leaves a sample only if the door is answered and there is a dog there.
P(answer)=3/4
P(dog) = 2/3
a) Determine the probability that Fred gives away his first sample on his third call.
b) Given that he has given away exactly four samples on his first eight calls, determine the conditional probability that he will give away his fifth sample on his eleventh call.
c) Given that he did not give away his second sample on his second call, determine the probability that he will leave his second sample on his fifth call.
d) Fred need s "a new supply" immediately after the call on which he gives away his last can. Fred starts with two cans. Determine the probability that he completes at least five calls before needing a new supply.
3. Consider the manufacture of Grandmother's Fudge Nut Butter Cookies. Grandmother has noted that the number of nuts in a cookie is a random variable with a Poission mass function and that the average number of nuts per cookie is 1.5.
a) What is the probability of having at least one nut in a randomly selected cookie?
b) What is the variance in the number of nuts per cookie?
c) Determine the probaility that a box of M cookies contains expected number of nuts in a box of M cookies.
d) What is the probability that a nut selected at random goes into a cookie containing K nuts?
4. MATLAB SIMULATION OF EM. Use the model of the world to test the EM algorithm (See notes on web page reading list).
a) Randomly generate from the two Gaussians and then see if the EM algorithm converges.
b) Test what happens when the world has one or three gaussians but the model thinks there are two.
5. Compare the classification results for your model to the nearest neighbor technique.