Problem Set 3 for ECE 313: Probability Theory | Assignments Statistics

University Problem Set #3 ECE 313

of Illinois Page 1 of 2 Spring 1999

Assigned: Wednesday, February 3, 1999

Due: Wednesday, February 10, 1999

Reading: Ross, Chapter 4.1, 4.3–4.5, 4.7, and Chapter 3

Noncredit Exercises: (Do not turn these in) Ross, pp. 173-184: 2, 7, 13, 20, 28, 35(a), 39,

40-43, 57, 59; pp. 184-188: 11, 13, 15-18

Problems:

1. The experiment consists of picking a student from the set of all UIUC students registered

this semester. It is not necessary to assume that all students are equally likely to be picked,

but you may make this assumption if it makes you feel happier and more confident.

(a) Let A and B denote the events that the student picked has had respectively four years of

science (FYS) and calculus in high school. Let P(A) = 0.45 and P(B) = 0.35. If the

probability that the student had neither FYS nor calculus is 0.3, what is the probability that

the student had both FYS and calculus? What is the probability that the student had FYS

but not calculus ?

(b) Let C denote the event that the student is registered in ECE 313, and let A and B be as in

part (a). Suppose that P(A∩B∩C)=0.002. What is the probability that the student picked

is not registered in ECE 313, but did have both FYS and calculus ? If the probability that

the student picked is registered in ECE 313, and has had either FYS or calculus (but not

both) is 0.004, and if students who had neither FYS nor calculus did not register in ECE

313, what is P(C)?

compute? It is not necessary to actually compute each probability.

P(A ∪ C), P(A∪B∪C), P(A ∪ B∪ Cc), P(AcBcCc), P(AcBCc), P(ABCc)

2. Find P(A ∪ (Bc ∪ Cc)c) in each of the following four cases:

(a) A, B, and C are mutually exclusive events and P(A) = 1/3.

(b) P(A) = 2P(BC) = 4P(ABC) = 1/2 (c) P(A) = 1/2, P(BC) = 1/3, and P(AC) = 0

(d) P(Ac ∩ (Bc ∪ Cc)) = 0.6

3. Use a spreadsheet/Mathematica/MATLAB for this problem.

Let A denote an event of probability p.

(a) For p = 0.1, 0.25. 0.4, 0.5, 0.6, 0.75, and 0.9, find the numerical values of the

probabilities that A occurs 0, 1, 2, … , 10 times on 10 trials of the experiment.

(b) You have, in effect, computed the probability mass function for a binomial random variable

X with parameters (10, p) for seven choices of p. For each value of p, draw a bar graph of

the pmf. (For p = 0.5, the answer is shown on page 151 of Ross!)

p = 0.25 and p = 0.75? for the cases p = 0.4 and 0.6?

(d) Prove mathematically that if X is a binomial random variable with parameters (n, p), then

Y = n – X is a binomial random variable with parameters (n, 1–p).

(e) From each of the seven graphs of part (b), find the value of k for which P{X = k} is

maximum. Compare your results to the prediction of Proposition 7.1, p. 150, of Ross.

4. Let X denote a binomial random variable with parameters (N, p). What is the probability

that X is an even integer? Remember that 0 is an even integer.

[Hint: What is (x+y)N + (x – y)N?]

5. There are N multiple-choice questions on a certain examination. A student knows the

answer to K of these and marks the answer sheet accordingly. For the remaining N – K

questions, the student guesses randomly among the five choices. The examiner can easily

determine C, the number of correct answers on the answer sheet, but is more interested in

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University Problem Set #3 ECE 313

of Illinois Page 1 of 2 Spring 1999

Assigned: Wednesday, February 3, 1999 Due: Wednesday, February 10, 1999 Reading: Ross, Chapter 4.1, 4.3–4.5, 4.7, and Chapter 3 Noncredit Exercises: (Do not turn these in) Ross, pp. 173-184: 2, 7, 13, 20, 28, 35(a), 39, 40-43, 57, 59; pp. 184-188: 11, 13, 15- Problems: 1. The experiment consists of picking a student from the set of all UIUC students registered this semester. It is not necessary to assume that all students are equally likely to be picked, but you may make this assumption if it makes you feel happier and more confident. (a) Let A and B denote the events that the student picked has had respectively four years of science (FYS) and calculus in high school. Let P(A) = 0.45 and P(B) = 0.35. If the probability that the student had neither FYS nor calculus is 0.3, what is the probability that the student had both FYS and calculus? What is the probability that the student had FYS but not calculus? (b) Let C denote the event that the student is registered in ECE 313, and let A and B be as in

part (a). Suppose that P(A∩B∩C)=0.002. What is the probability that the student picked is not registered in ECE 313, but did have both FYS and calculus? If the probability that the student picked is registered in ECE 313, and has had either FYS or calculus (but not both) is 0.004, and if students who had neither FYS nor calculus did not register in ECE 313, what is P(C)? (c) Using the data given in parts (a) and (b), which of the following probabilities can you compute? It is not necessary to actually compute each probability. P(A ∪ C), P(A∪B∪C), P(A ∪ B∪ Cc), P(AcBcCc), P(AcBCc), P(ABCc)

2. Find P(A ∪ (Bc^ ∪ Cc)c) in each of the following four cases: (a) A, B, and C are mutually exclusive events and P(A) = 1/3. (b) P(A) = 2P(BC) = 4P(ABC) = 1/2 (c) P(A) = 1/2, P(BC) = 1/3, and P(AC) = 0

(d) P(Ac^ ∩ (Bc^ ∪ Cc)) = 0.

3. Use a spreadsheet/Mathematica/MATLAB for this problem. Let A denote an event of probability p. (a) For p = 0.1, 0.25. 0.4, 0.5, 0.6, 0.75, and 0.9, find the numerical values of the probabilities that A occurs 0, 1, 2, … , 10 times on 10 trials of the experiment. (b) You have, in effect, computed the probability mass function for a binomial random variable X with parameters (10, p) for seven choices of p. For each value of p, draw a bar graph of the pmf. (For p = 0.5, the answer is shown on page 151 of Ross!) (c) What is the relationship between the pmfs for the cases p = 0.1 and p = 0.9? for the cases p = 0.25 and p = 0.75? for the cases p = 0.4 and 0.6? (d) Prove mathematically that if X is a binomial random variable with parameters (n, p), then Y = n – X is a binomial random variable with parameters (n, 1–p). (e) From each of the seven graphs of part (b), find the value of k for which P{ X = k} is maximum. Compare your results to the prediction of Proposition 7.1, p. 150, of Ross.

4. Let X denote a binomial random variable with parameters (N, p). What is the probability that X is an even integer? Remember that 0 is an even integer. [Hint: What is (x+y)N^ + (x – y)N?]

5. There are N multiple-choice questions on a certain examination. A student knows the answer to K of these and marks the answer sheet accordingly. For the remaining N – K questions, the student guesses randomly among the five choices. The examiner can easily determine C, the number of correct answers on the answer sheet, but is more interested in

University Problem Set #3 ECE 313

of Illinois Page 2 of 2 Spring 1999

estimating the value of K, since K is a better measure of the student’s knowledge than C. (Educators like to nitpick about such subtle differences!). Note that the number of w rong answers can be modeled as a binomial random variable W with parameters (N – K, 0.8). (a) The examiner notes that n questions have been answered incorrectly by the student, i.e. the event { W = n} is observed. Write an expression for P{ W = n} in terms of N, K, and n. (b) Obviously, 0 ≤ K ≤ N–n. Now, use the method used in the proof of Proposition 7.1, p. 150 of Ross to show that of all possible assumptions K = 0, K = 1, K = 2, … , K = N–n that the examiner might make, the assumption that K is the largest integer not exceeding N – 1.25n + 1, i.e. estimating K as K =^ N – 1.25n + 1 maximizes P{ W = n}. K is the^ called the maximum-likelihood estimate of K. Find the numerical value of K for the case^ N = 100 and n = 8. (c) Since C = N – n, examiners generally subtract one-fourth of the wrong answers from C

and estimate the value of K as K = C –~ 0.25n ≈ N – 1.25n. This is called applying the guessing penalty, and it can hurt scores slightly in the sense that the examiner’s estimate K~ might be smaller than the maximum-likelihood estimate K found in part (b).^ Compare K and ^ K for the case N = 100 and n = 8.~ Compare K and ^ K for the case N = 100 and n = 10.~ (d) If N = 100 and K = 90, which of the events { W = 0}, { W = 1}, … , { W = 10} has the largest probability? (Hint: See Proposition 7.1, p. 150 of Ross). Suppose that this largest probability event actually occurred. Does the examiner’s estimate K correctly estimate K?~ Does the maximum-likelihood estimate K correctly estimate K?^ (e) Continuing to assume that N = 100 and K = 90, what happens if by sheer dumb luck the student manages to guess right on 6 problems so that the event { W = 4} occurs? Compare this to the case when the event { W = 10} occurs (as in part (c)). (f) Continuing to assume that N = 100 and K = 90, find the probabilities that the examiner’s

estimate K respectively overestimates, underestimates, and correctly estimates K.^ (g) Noncredit Optional Exercise: Suppose that for each of the 10 questions to which the answer is not known to the student, the student can nonetheless correctly eliminate three answers as being obviously wrong. The student then chooses at random between the other two answers. Which of the events { W = 0}, { W = 1}, … , { W = 10} is the most probable? If this is the event which actually occurs, what is the examiner’s estimate K?^ (Note that this, in essence, gives some “partial credit” by rewarding the student for the partial knowledge that three of the five answers to each problem are wrong.) (g) Noncredit Optional Exercise: Write a 500-word essay on why it is more important to be lucky than smart.

6. Let X denote a Poisson random variable with unknown parameter λ. Suppose that the event { X = k} occurs.

(a) What is the maximum–likelihood estimate of λ? That is, what value of λ maximizes the probability of the observed event { X = k}? (b) Consider a binomial random variable Y with parameters (N, p) where the parameter p is unknown. If the event { Y = k} is observed (e.g. heads occurs k times on N tosses of a biased coin with P(Heads) = p), then we showed in class that ^p = k/N is the maximum– likelihood estimate of p. Since for large N and small p, the binomial random variable Y can be approximated by a Poisson random variable X with parameter λ = Np, it would seem reasonable that the maximum-likelihood estimate of λ would be ^λ = N^p = k. Does your answer to part (a) give this result?

Problem Set 3 for ECE 313: Probability Theory, Assignments of Statistics

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