1 Rules

1. You may use any books, notes, and internet sites.

2. You may use calculators and SW like Matlab or Mathematica.

3. You may not ask anyone for help.

4. You may not communicate with anyone else about the course or the exam.

5. You may not accept help from other people. E.g., if someone offers to give you help w/o your asking, you may not accept.

6. You have 24 hours.

7. That is, your answers must be in gradescope by 4pm (1600) EDT Tues.

8. Email me with any questions. Do not wait until just before the due time.

9. Write your answers on blank sheets of paper and scan them, or use a notepad or app to write them directly into a file. Upload it to gradescope.

10. You may mark any 10 points as FREE and get the points.

11. Print your name and rcsid at the top.

2 Questions

1. Consider this pdf:

   $f_{X,Y} (x,y) = c (x^2 + 2 x y + y ^2) $ for $ 0\le x,y \le 1 $, 0 otherwise.

   1. (5 points) What must $c$ be?

   2. (5) What is F(x,y)?

   3. (5) What is the marginal $f_X(x)$?

   4. (5) What is the marginal $f_Y(y)$?

   5. (5) Are X and Y independent? Justify your answer.

   6. (30) What are $E[X], E[X^2], VAR[X], E[Y], E[Y^2], VAR[Y]$ ?

   7. (15) What are $E[XY], COV[X,Y], \rho_{X,Y}$ ?

2. (5) This question is about how late a student can sleep in before class. He can take a free bus, if he gets up in time. Otherwise, he must take a $10 Uber.

   The bus arrival time is not predictable but is uniform in [9:00, 9:20]. What's the latest time that the student can arrive at the bus stop and have his expected cost be no more than $5?

3. (5) X is a random variable (r.v.) that is U[0,1], i.e., uniform [0,1]. Y is a r.v. that is U[0,X]. What is $f_Y(y)$ ?

4. (5) X is an r.v. U[0,y] but we don't know y. We observe one sample $x_1$. What is maximum likelihood for y?

5. This is a noisy transmission question. X is the transmitted signal. It is 0 or 1. P[X=0] = 2/3. N is the noise. It is Gaussian with mean 0 and sigma 1.

   Y = X + N

   1. (5) Compute P[X=0|Y].

   2. (5) Compute $g_{MAP}(Y)$.

6. Let X be a Gaussian r.v. with mean 5 and sigma 10. Let Y be an independent exponential r.v. with lambda 3. Let Z be an independent continuous uniform r.v. in the interval [-1,1].

   1. (5) Compute E[X+Y+Z].

   2. (5) Compute VAR[X+Y+Z].

7. (5) We have a Gaussian r.v. with unknown mean $\mu$ and known $\sigma = 100$. We take a sample of 100 observations. The mean of that sample is 100. Compute $a$ such that with probability .68, $100-a \le \mu \le 100+a$.

8. (5) You're testing whether a new drug works. You will give 100 sick patients the drug and another 100 a placebo. The random variable X will be the number of days until their temperature drops to normal. You don't know in advance what $\sigma_X$ is. The question is whether E[X] over the patients with the drug is significantly different from E[X] over the patients with the placebo.

   What's the best statistical test to use?

9. You're tossing 1000 paper airplanes off the roof of the JEC onto the field, trying to hit a 1m square target. The airplanes are independent. The probability of any particular airplane hitting the target is 0.1%. The random variable X is the number of airplanes hitting the target.

   1. (5) What's the best probability distribution for X?

   2. (5) Name another distribution that would work if you computed with very large numbers.

   3. (5) Name another distribution that does not work in this case, but would work if the probability of any particular airplane hitting the target is 10%

      Historical note: for many years, GM week had an egg toss. Students designed a protective packaging for an egg and tossed it off the JEC onto the brick patio. Points were given for the egg surviving and landing near the target.

   4. You want to test a suspect die by tossing it 100 times. The number of times that each face from 1 to 6 shows is this: 12, 20, 15, 18, 15, 20.

      1. (5) What's the appropriate distribution?

      2. (5) If the die is fair, what's the probability that the observed distribution could be that far from the actual probability?

Total: 140 points.