1. Three Basic Ingredients of Probability Theory
2. Sets and Set Operations; Events Text 2.1, 2.2
3. Counting 2.3
4. Counting, Probabilities of Events 2.3, 2.4
5. Additive Rules 2.5
6. Conditional Probabilities and Independence 2.6
7. Multiplicative Rules and Bayes' Theorem 2.7, 2.8
8. Concept of a Random Variable: Examples 3.1
9. Discrete Random Variables: PDF and CDF 3.2
10. Continuous Random Variables: PDF and CDF 3.3
11. Discrete and Continuous Random Variables Continued
12. Examples: Binomial for small n; Exponential
13. Review
14. Exam 1 (Lectures 1-13) Fri. Sept. 24
15. Joint Probability Distributions: Discrete 3.4
16. Joint Probability Distributions: Continuous 3.4
17. Mean or Expectation of a Random Variable 4.1
18. Variance and Covariance 4.2
19. Means and Variances of Linear Combinations of Random Variables 4.3
20. Uniform Discrete Distribution 5.1,5.2
21. Binomial Distribution: General Case 5.3
22. Hypergeometric Distribution 5.4
23. Geometric and Negative Binomial Distributions 5.5
24. Poisson Distribution and the Poisson Process 5.6
25. Review
26. Exam 2 (Lectures 15-25) Mon. Oct. 25
27. Continuous Uniform Distribution 6.1
28. Normal Distribution 6.2
29. Areas Under the Normal Curve 6.3
30. Applications of the Normal Distribution 6.4
31. Gamma and Exponential Distributions 6.6,6.7
32. Chi-Squared Distribution 6.8
33. Functions of Random Variables 7.1,7.2
34. Functions of Random Variables Continued
35. Moments and Moment Generating Functions 7.3
36. Moments and Moment Generating Functions 7.3
37. Review
38. Exam 3 (Lectures 27-37) Mon. Nov. 22
39. Sums of Random Variables Notes
40. The Central Limit Theorem 8.5; Notes
41. Random Sampling, Some Important Statistics 8.1,8.2
42. Sampling Distributions and Sampling Distributions of Means 8.4,8.5
43. Review
44. Review
P29 4,5,9,10,17,19,20 P38 2,5,12,14,16,18,20,29 S 1,2,3 (S = Special Problems are listed below)
HW2 (Lectures 5-7) Due 9/13
P46 1,6,9,14 P54 2,7,10,16,22,23 P60 1,3,6,7,8 S 4,5,6
HW3 (Lectures 8-12) Due 9/22
P73 2,3,4,6,7,9,11,13,14,18,24 S 7,8,9
HW4 (Lectures 15,16) Due 10/4
P84 1,2,6,8,13 P86 3,4,9
HW5 (Lectures 17-19) Due 10/11
P94 1,2,4,5,16,19 P102 2,3,4,11,13 P112 1,6,7,8,14,15
HW6 (Lectures 20-22) Due 10/18
P124 1,3,5,12,16,22,25,28 P131 4,8,16,20 S 10
HW7 (Lectures 23-24) Due 10/22
P139 3,4,7,12 P140 (Review Exercises) 6,12 S 11,12,13,14
HW8 (Lectures 27-30) Due 11/05
P156 1,2,3,4,9,13 S 14,15,16
HW9 (Lectures 31-34) Due 11/12
P174 6,8,14,15 P175 (Review Exercises) 2,8,13,16 P191 2,5,7 S 17
HW10 (Lectures 35,36) Due 11/19
P191 1,2,13,16,17,18,19,20,21
HW11 (Lectures 39,40) Due 12/03
P215 5,8,10,11 S 18
HW12 (Lectures 41,42) Due 12/08
P200 3,11,12,13,14 P215 1,2,3,12
SPECIAL PROBLEMS
1. A radar system gives range and elevation of aircraft. (Elevation is an angle.) If the
experiment consists if reading target range and elevation of any incoming aircraft, describe
the sample space S for this experiment, and draw a diagram of it.
2. A communication system sends binary messages of length 6 (e.g. 011101), i.e. 6 bits.
(a) How many different messages are possible?
(b) How many have exactly 3 zeros?
(c) How many have more zeros than ones?
3. How many distinct 8-bit strings are there if
(a) the first and last bits must be identical?
(b) the last 2 bits cannot be 00 ?
(c) the string cannot contain 111 (i.e. 3 consecutive ones)?
4. A binary communication system sends zeros with probability .5 and they are
correctly received with probability .95 ; ones are sent with probability .5 and
correctly received with probability .97.
(a) What is the probability of an error: i.e. a bit is sent and the other bit is received?
(b) If a zero is received, what is the probability a zero was sent?
5. A binary communication system sends zeros with probability .6 and ones with
probability .4 . Suppose a zero has probability 1 - a of being received correctly,
and 1 - a/2 for a one. What must a be for the total probability of an
error to be less than .001 ?
6. A communication network (see Figure) has links which fail independently of one
another with probability p . If there are one or more connected paths from LA
to Boston, the network is up.
(a) What is the probability the network is down? (This is a function of p .)
(b) What must p be so that the network is up with probability .999 ?
7. You are buying CD's at Big Bob's Binary Bargain Barn. They are defective
with probability 1/8 .
(a) You buy 5. What is the probability they are all good?
(b) You buy 3. What is the probability at least one is defective?
8. A random variable V representing a voltage has PDF given by
f(v) = v for 0 < v < 1 and f(v) = 2-v for 1< v < 2.
(a) What is the probability that the voltage is between 0 and 1/3 ?
(b) Is V a discrete or continuous random variable? Explain.
(c) What is the CDF for the voltage? Sketch its graph.
9. Consider the random variable V representing voltage from 8. above. Suppose
the voltage is run through an analog to digital converter with 6 possible results:
1/6 if between 0 and 1/3 (including 1/3) ; 1/2 if between 1/3 and 2/3 (including
2/3) ; ... , and 11/6 if between 5/3 and 2 (including 2).
(a) Find and sketch the graph of the PDF for the converted (digital) voltage.
(b) Find and sketch the graph of the CDF for the converted (digital) voltage.
(c) Use the CDF to compute P(1/2) for the converted (digital) voltage.
(d) Is the converted (digital) voltage a discrete or continuous random variable? Explain.
10. An 8 bit binary message is sent. The probability each bit is received
correctly is .90 , and they are independent of each other.
(a) What is the probability the whole message is correctly received?
(b) What is the probability that exactly 2 of the bits are errors?
(c) A 9th "check" bit is sent which is zero if the sum of the first 8 bits is even and
1 if it is odd. An error is detected if the received check bit and the 8 bit sum don't
match - e.g. the received sum is odd but the received check bit is zero. What is
the probability exactly 1 of the 8 bits will be in error and will also be detected?
11. A 12 bit message is sent. Bit errors occur independently with probability .10 .
What is the probability that
(a) less than 3 errors will be made?
(b) the first error occurs within the first 4 bits?
(c) the second error occurs in the ninth bit?
12. A message is sent over a network and rejected if the receiver is busy. It is
repeatedly retransmitted until it is successfully received. Suppose the receiver is
busy with probability 1/6 . What is the probability that
(a) a message is received in 3 or less tries?
(b) 3 messages will require exactly 4 tries?
(c) 4 different messages will take a total of 5 or fewer tries?
13. On average, 100 packets per minute arrive at a particular web server.
Assuming packet arrival is a Poisson process, find the probability that
(a) no packets arrive in the next 2 seconds?
(b) 3 or more arrive in the next 4 seconds?
(c) less than 250 arrive in the next 3 minutes?
14. Photodetectors produce a current pulse when a photoelectron hits
a detector. There is some noise in these devices, one component of which
is Poisson (called "shot noise") - that is, random photoelectrons hit the
detector in a Poisson fashion and produce corresponding current pulses.
Suppose the shot noise pulses occur on average twice per millisecond.
(a) What is the probability there will be more than 3000 in the next second?
(b) There are cases where the total shot noise current in the photodetector
at any one time can be shown to be proportional to the average rate of
the Poisson events: i.e. I (total, shot) = a l , where a is a proportionality
constant. If, in the above example, this I = 2 milliamps, what is a, and what
are its units?
15. (Signal Detection) Suppose that the two signals 0 and 1 are represented
by voltages ( in order to send them) and they are equally likely to be sent.
Now the voltage (in millivolts) for the signal 0 is a normal random variable with
mean 5 and standard deviation 3 ; for the signal 1 the mean is 8 and standard
deviation 2. To decide whether we have received a 0 or 1, we pick a voltage
value c and register a 0 if the received signal is below c, and 1 otherwise.
(a) Find the probability a 0 is sent and a 1 is received. (This is a function of c.)
(b) Same for 1 sent and 0 received.
(c) Find the total probability of an error as a function of c.
(d) Find the value of c that minimizes the total error, and prove it works.
(e) Redo (a)-(d) assuming 0 is sent with probability .6 and 1 with probability .4 .
16. We are reading voltages with a voltmeter. The meter's detection errors are
normally distributed with mean 0 volts and standard deviation 2 millivolts. We need
to sum the first 100 voltages we read. What is the probability the sum voltage
will be off by
(a) more than 15 millivolts ?
(b) less than 5 millivolts ?
17. The noise X in a particular electronic system is normally distributed with
mean 0 and standard deviation s . The normalized instantaneous power of
the noise Y is given by Y = X^2/s ^2 . Find the PDF of Y.
18. A set of batteries for your CD lasts you an average of 2 weeks with a
standard deviation of 3 days.
(a) What is the approximate probability that you will need more than
24 sets of batteries in the next year?
(b) Less than 30 sets?
(c) You are 90% sure you will need at most how many sets?