1.   Your best friend is a first year student in med school at Johns Hopkins.  He calls and tells you that he has failed his first anatomy exam.   The good news is that he can retake the    exam until he passes it.  The bad news is that no matter how much he studies, he has a     probability of passing the exam that remains at a fixed 70%.  (7 pts)

a.   Which distribution would correctly model the number of times your friend will have to take this exam before he passes?

b.   What is the probability that your friend will fail the exam 4 times before passing?

2.   You are going to take a vacation each year for the next 10 years.  Each year, you have the same probability of choosing a vacation on a particular continent as you did in the year    before.  These probabilities are listed in the table below. (7 pts)

a.   Which distribution would correctly model the likelihood of going to each continent a particular number of times in the next 10 years?

b.   What is the probability that, of your 10 vacations, 4 are in North America, 2 are in South America, 3 are in Europe and 1 is in Asia?

3.   Bobby and Francine are twins, and for their birthday they are each going to get a puppy. They go to the local rescue, and look at puppies one at a time.  Both children have to like a puppy in order to take it home; they will select 2 puppies total; the individual                 probability that the twins like a particular puppy is 65% and is independent of the            likelihood that they like any other puppy. (7 pts)

a.   What is the appropriate distribution to model the number of puppies that the children will see before selecting their puppies and leaving the rescue?

b.   What is the probability that the 15th puppy is the second one they choose to take home?

4.   A cheesecake sampler has 30 cheesecake squares – 5 pieces each of 6 flavors.  Anant   buys 1 sampler for a party, even though he only likes two of the cheesecake flavors.  (7 pts)

a.   What is the appropriate distribution to model the probability that Anant likes a    given number of cheesecake squares if he eats a fixed number from the sampler?

b.   If Anant picks 4 cheesecake squares to eat, what is the probability that he does not like any of them?

5.   Students arrive to class according to a Poisson process with λ= 5 arrivals/minute.  ( 15 pts)

a.   What is the probability that at least 2 students arrive in 1 minute?

b.   What is the probability that exactly 8 students arrive in 2 minutes?

c.   What is the distribution (with relevant parameter) of the interarrival times?

6.   The amount of time Frank spends walking around Disney World looking for a place to buy a churro follows an exponential distribution with a mean of 10 minutes.  (10 pts)

a.   Given that Frank has been walking around looking for a place to buy a churro for

5 minutes already, what is the probability that he will search for at least 15 minutes total?

b.   What property of the exponential distribution makes this problem easy to solve?

7.   Answer the following questions about the normal distribution. (27 pts)

a.   You are conducting an experiment and have 4 data points so far.  Even though   this is a very small sample, you want to know if your data up to this point is       approximately normally distributed, so you are going to make a qq plot. What is the value for the quantile of the normal distribution that you will use to generate your qq plot corresponding to the third data point?

b.   Let Z be a random variable following a standard normal distribution.

i.   What is Pr(Z ≤ 0)?

ii.   What is Pr(Z ≥ 6)?

c.   Let X be a normally distributed random variable with mean 60 and variance 36. What is Pr(48 ≤ X ≤ 72)?

d.   Let Y be a discrete random variable for which you have a normal approximation with mean 60.5 and variance 36.  Use this approximation to estimate Pr(Y≤ 66).

e.   What are the conditions that have to hold to be able to use the normal approximation of the binomial distribution?

8.   Answer the following questions about specific distributions. (25 pts)

a.   Use the Gamma Distribution table to find F(18; 5, 6).  You may choose to handle the integral instead, but it is more time consuming.

b.   Without taking an integral – evaluate the gamma function for alpha = 6.

c.   What is the relationship between Bernoulli and the Binomial random variables?

d.   Using the specific case of n = 2 and p = 0.4, show that the expectation of a binomial random variable is np.

e.   Evaluate the integral to derive the CDF of the exponential distribution from the pdf.