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EE 104 - Probabilistic Systems Analysis

Problem Set #3

1. A deck of cards has 2 joker cards in addition to the normal 52 cards. You are dealt a five-card hand.

(a) What is the probability of getting neither joker?

(b) What is the probability of getting one of the jokers?

(c) What is the probability of getting both jokers?

2. A data packet consists of 150 bits, and is sent over a data channel with a bit error probability of 0.001.

(a) What is the probability of successful packet transmission with no bit errors?

(b) In order to compensate for this noisy channel, some of the bits are used to do error correction coding, which builds some redundancy into the message. Suppose the code is able to catch and correct single-bit errors and two-bit errors. What is the probability of successful packet transmission?

3. This problem introduces the concept of a receiver operating characteristic (sometimes called a ROC curve). An underwater sonar sends out pulses of acoustic energy and tries to detect pulses coming back after bouncing off targets (sharks, underwater mines, submarines). But the problem is how to be sensitive even to weak returns (have high probability of detection) without reporting false detections from environmental noise or electrical noise in the detection circuit (low probability of false alarm). The ROC curve shows the tradeoff between these performance metrics. The receiver circuit uses a comparator circuit: if the detector signal exceeds some threshold voltage v, it declares a detection, i.e., it “claims” there’s a target. By adjusting v to a low value, we can detect weak pulses, but we also get more false alarms. If v is high, many genuine returns are too weak to trigger the comparator and we miss them. Suppose we have a system that detects a fraction pdet = exp(-v) of the genuine returns from a typical target, but, in the absence of a target produces a false alarm on each pulse with a probability pfa = exp(-5v). The figures below show this single-pulse performance. The left hand plot shows pdet and pfa as functions of the threshold voltage v. The right hand plot shows the ROC curve, which is a parametric plot of pdet(v) vs 1- pfa(v). Ideally, we’d like to get up to the smiley face, where we detect the target all the time but never get a false alarm when the target is absent.

(a) Using MATLAB, Mathematica or your favorite tool, reproduce the ROC curve on the right side of the above figure.

(b) Now let’s reach for the smiley face. Instead of basing our decision on a single pulse, let’s send out three pulses, and say that if we get return pulses bigger than v on at least 2 out of the 3, we declare a real target. One or less, we don’t. Derive expressions for the false alarm probability (declaring a target when there is none) and for the detection probability (declaring a target when one is present) based on this two-thirds voting scheme.

(c) On the same graph, plot the ROC curve from part a and the improved one from part b.

ROC curves are useful in many signal detection contexts. One important example is testing for a disease, where you want catch everyone who has it but also avoid false positives.

4. The DNA molecule consists of twisted strands constructed from an ordered sequence of “base pairs” that come in four “flavors” abbreviated as A, C, G, and T. If you have a strand of 15 base pairs, how many ways can it consist of three each of A, C, and G and six Ts?

5. In a certain town, 20% of the eligible voters prefer candidate A, 10% prefer candidate B, and the remaining 70% have no preference. You randomly poll 10 voters. What is the probability that you’ll get the “average result” that 2 will prefer candidate A, 1 will prefer candidate B, and the remaining 7 will have no preference?

6. A random variable has a probability mass function: PN(n) = A exp(-2n) for n = 0,1,2,3…. and = 0 otherwise.

(a) What must the constant A be for this to be a valid PMF?

(b) What is the probability that n ≤ 2?

(c) This is what type of random variable? (i.e., Pascal, Bernoulli, Binomial?)

7. You take an exam that contains 15 multiple-choice questions. Each question has 4 choices. You know the answer to 10 questions, but you have no idea about the other 5 questions so you choose answers randomly. Your score x on the exam is the total number of correct answers.

(a) Find the PMF of x and plot it as a bar chart.

(b) The number you get right on the guessed questions is what kind of random variable?

8. Computer data networks have used different ways of sharing a common communication line to avoid the problem of collisions, where more than one computer sends data simultaneously and this causes the messages to be garbled. For example, the computers could take turns transmitting by passing around a token, an electronic version of the Native American talking stick. But this wastes time giving turns to computers that have nothing to send. The ALOHA protocol just “plays casino” with the problem. Any computer with a data packet to send just starts sending; if during this transmission it detects data coming from another computer, it stops transmitting, waits a random amount of time, and then tries transmitting again.

(a) Suppose your computer has a probability p of being able to transmit a packet, and a probability 1-p of getting interrupted by a collision. What kind of random variable is n, the number of transmission attempts needed to successfully transmit the packet?

(b) Sketch a labelled bar chart of the PMF of n for a collision probability of 80%

(c) Plot the probability of successfully transmitting the packet in at most two attempts as a function of the collision probability.

ALOHA was the basis for the Ethernet.

9. As a check on the quality of an electronics assembly line, a quality control engineer gives a pass/fail test to parts coming off the end of the assembly line until she is able to collect 12 good parts. She then reports the number z, which is the number of parts she had to test to get the 12 good ones.

(a) What kind of random variable is z?

(b) How high must the probability of a part passing be in order for her to have a 35% chance of getting 12 good parts from the first 12 she tests?