.. title: Engineering Probability Class 26 and Final Exam Mon 2020-04-27
.. slug: final-2020
.. date: 2021-05-08
.. tags: class
.. link: 
.. description: 
.. type: text
.. has_math: true

.. sectnum::
.. contents:: Table of contents::
..


Rules
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a. You may use any books, notes, and internet sites.
#. You may use calculators and SW like Matlab or Mathematica.
#. You may not ask anyone for help.
#. You may not communicate with anyone else about the course or the exam.
#. 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.
#. You have 24 hours.
#. That is, your answers must be in gradescope by 4pm (1600) EDT Tues.
#. Email me with any questions.  Do not wait until just before the due time.
#. 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.
#. You may mark any 10 points as FREE and get the points.
#. Print your name and rcsid at the top.


Questions
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#. 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.

   a. *(5 points)* What must $c$ be?

   #. *(5)* What is F(x,y)?

   #. *(5)* What is the marginal $f_X(x)$?
      
   #. *(5)* What is the marginal $f_Y(y)$?

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

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

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

#. *(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?

#. *(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 $f_Y(y)$ ?

#. *(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?

#. 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

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

   #. *(5)* Compute $g_{MAP}(Y)$.

#. 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].

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

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

#. *(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,  $100-a \\le \\mu \\le 100+a$.

#. *(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?
   
#. 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.

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

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

   #. *(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.*

   #. 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.

      a. *(5)* What's the appropriate distribution?

      #. *(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.