# tutorials + solutions

of 21/21
Department of Electrical and Electronic Engineering Mr R A JUGURNAUTH PROBABILITY PROBLEM SHEET ONE: Basic Statistics, Sets and Combinations Question 1 Find the mean ( x μ = ), mean square value ( 2 x ), the variance ( 2 σ ), and the standard deviation ( σ ) for the following data: { } . Confirm that 1 0 {1, 6, 9, 4, 2, 6, 5, 8} N i i x = = 2 2 x 2 σ μ = + for this particular case. Show that 2 2 x 2 σ μ = + holds for any set of data. Finally, a standard deviation of 2 about a mean of 4 represents a different effect than a standard deviation of 2 about a mean of 20. So we define the coefficient of variation as 100% V σ μ = × . What is V in this case? Does all the data lie less that two standard deviations from the mean? Question 2 Standard units tell the number of standard deviations a given value lies above or below the mean of a population/data set. Let the value of x in standard units, denoted by z, be defined as: x z μ σ = . It can be used to compare values from different samples or populations. Ten students take two tests - Maths and Physics. In the Maths test Mary gets 67% and 60% in the Physics test. Given that one test may be considered ‘harder’ than the other, use standard units to determine in which test did Mary get a ‘higher’ score. The % scores for the ten students for the two tests is given below. Maths Marks: {67 74 44 93 46 41 84 52 20 1} Physics Marks: {60 35 81 0 13 20 19 5 27 19} Question 3 Which set of data goes with which histogram? The means are 0.536 and 4.096; the standard deviations are 0.2630 and 1.728. Which goes with which? 0 10 20 30 40 50 60 70 80 90 100 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 2 4 6 8 10 12 HISTOGRAM OF X 1 2 3 4 5 6 7 8 9 D1 -1 0 1 2 3 4 5 6 7 8 9 0 5 10 15 20 25 HISTOGRAM OF X H2 H1 D2 Question 4 Let { } 1, 2, 3, 4, U be the universal set and let = K { } { } { } { 1, 2, 3, 4 , 3, 4, 5, 6, 7 , 6,7,8,9 and 2, 4, 6, A B C E = = = = K } . Find the following sets: (i) , , and A B E ; (ii) \ , \ , \ , \ and \ A BBCB AC B C E ; (iii) ; and A B B C (iv) From the above results confirm that ( \ ) ( \ ) and ( \ ) ( \ ) A B A B B A B C B C C B = = . 1

Post on 08-Nov-2014

371 views

Category:

## Documents

Tags:

• #### bond strength

Embed Size (px)

DESCRIPTION

solutions for university of Mauritius exams for analytical techniques

TRANSCRIPT

Department of Electrical and Electronic Engineering

Mr R A JUGURNAUTH

PROBABILITY PROBLEM SHEET ONE: Basic Statistics, Sets and CombinationsQuestion 1 Find the mean ( = x ), mean square value ( x 2 ), the variance ( 2 ), and the standard deviation ( ) for the following data:1 = {1, 6,9, 4, 2, 6, 5,8} . { xi }iN =0

Confirm that x 2 = 2 + 2 for this particular case. Show that

x 2 = 2 + 2 holds for any set of data. Finally, a standard deviation of 2 about a mean of 4 represents a different effect than a standard deviation of 2 about a mean of 20. So we define the coefficient of variation as V = 100% . What is V in this case? Does all the data lie less that two standard deviations from the mean? Question 2 Standard units tell the number of standard deviations a given value lies above or below the mean of a x population/data set. Let the value of x in standard units, denoted by z, be defined as: z = . It can be used to compare values from different samples or populations.Ten students take two tests - Maths and Physics. In the Maths test Mary gets 67% and 60% in the Physics test. Given that one test may be considered harder than the other, use standard units to determine in which test did Mary get a higher score. The % scores for the ten students for the two tests is given below. Maths Marks: Physics Marks: Question 3 {67 {60 74 35 44 81 93 0 46 13 41 20 84 19 52 5 20 27 1} 19}

D1

HISTOGRA M OF X

9 8

12

H1

10

7 68

5 4 34 6

2 1 0 -1 0 10 20 30 40 50 60 70 80 90 100

2

HIS TOGRA M OF X 250 0 0.9 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

1

20

0.8 0.7

H2

15

0.6 0.5

D2

10

0.4 0.3

5

0.2 0.1

0 -1

0

1

2

3

4

5

6

7

8

9

0 0 10 20 30 40 50 60 70 80 90 100

Which set of data goes with which histogram? The means are 0.536 and 4.096; the standard deviations are 0.2630 and 1.728. Which goes with which? Question 4 Let U = {1, 2,3, 4,K} be the universal set and let A = {1, 2,3, 4} , B = {3, 4,5, 6, 7} , C = {6, 7,8,9} and E = {2, 4, 6,K} . Find the following sets: (i) A, B , and E ; (ii) A \ B, B \ C , B \ A, C \ B and C \ E ; (iii) A B and B C ; (iv) From the above results confirm that A B = ( A \ B) ( B \ A) and B C = ( B \ C ) (C \ B) .

1

Question 5 Let U = {a, b, c, d , e} be the universal set and let A = {a, b, d } , B = {b, d , e} . Find the following sets:

A B ; B A; B; B\ A; (v) A B ; (vi) A B ; (vii) A B ; (viii) B \ A ; (ix) ( A B) ;(i) (ii) (iii) (iv) (x) ( A B) . Question 6 (i) Using a Venn diagram, prove the OR rule for probabilities for two events i.e. prove P ( A B ) = P ( A) + P ( B) P( A B) , which is the same as P( A or B ) = P( A) + P( B ) P( A and B) ; (ii) Using a Venn diagram prove the OR rule for probabilities for three events i.e. prove P( A B C ) = P( A) + P( B) + P(C ) ( P( A B) + P( A C ) + P( B C ) ) + P( A B C ) ; (iii) What is the general rule for n events? (iv) Using the above results, what is the probability that a bridge hand (13 cards) contains 4 aces or 4 kings? (v) Using the above results, what is the probability that a poker hand (5 cards) contains all spades or all hearts? Question 7 I deal a hand of 5 cards from 52 playing cards. (i) What is the probability that all cards are clubs? (ii) What is the probability that at least one is a King? (iii) What is the probability that we have two pairs? Question 8 Consider the following experiments, and answer the questions. Experiment One: Select a ball from an urn containing balls numbered 1 to 50. Experiment Two: Pick a number at random between zero and one Experiment Three: Pick two numbers X and Y at random between zero and one. Experiment Four: Pick a number X at random between zero and one, then pick a number Y at random between zero and X. Experiment Five: Select a ball from an urn containing balls numbered 1 to 4. Suppose that balls 1 and 2 are black and that balls 3 and 4 are white. Note the number and the colour of the ball that you select. (i) Give expressions for the sample spaces for the first five experiments. (ii) Draw the sample spaces for the first four experiments. (iii) Give expressions for the following events in experiment one: (a) E1 = Even ball selected; b) E2 = Odd ball selected; c) E3 = Prime number ball selected; d) Give some relationships between the sets E1 , E2 and E3. (iv) Consider experiment four. Sketch the event E4 where the sum of X plus Y is greater than 0.5. Question 9 (i) A bag contains one red and three white balls. If two are picked at random, what is the probability that both are white? Answer this question in four different ways using either an exhaustive proof; combinations; permutations; or individual probabilities. (ii) Repeat four proofs for one red and three whites. Choose three. What is probability that two are white? Question 10 A bag contains 3 white balls, 4 black balls and 5 red balls. If I choose four balls at random find the probability that (i) I have zero reds; (ii) I have one red; (iii) I have two reds; (iv) I have at least three reds; (v) Repeat (iv) a different way. Question 11 Seven husbands (H1 to H7) are separated from their seven wives (W1to W7). They are then paired at random. (i) What is the probability that H1 is matched with his wife?; (ii) What is the probability that H2, H5 and H7 are all correctly matched?; (iii) What is the probability that at least one husband is matched to his wife? (iv) What is the probability that no husband is matched to his wife?

2

Department of Electrical and Electronic Engineering

Mr R A JUGURNAUTH

PROBABILITY PROBLEM SHEET TWO: Conditional and Bayesian Probability

Question 1 Consider three mutually exclusive and exhaustive events: A1 , A2 and A3 . Let there be another event, B . With the aid of Venn diagrams, give a simple an intuitive derivation of Bayes Law to expand P ( A1 | B ) . Also explain the purpose of Bayes Law. Question 2 If two events are such that the occurrence (or non-occurrence) of one, does not affect the likelihood of the other event, the events are called independent. The mathematical definition is: A and B are independent if P ( A B ) = P ( A) P ( B ) What does this say about the conditional probabilities? Question 3 (i) Prove that P ( A | B ) = 1 P ( A | B ) (ii) Prove that P ( A B C ) = P ( A) P ( B | A) P (C | ( A B) (iii) Generalise (ii) for P( A B C D ) , and others. Question 4 Find the probability that a poker hand (of five cards) contains two Kings if you know that it already contains exactly one Queen. Question 5 Suppose a box contains five white and seven red balls. Two balls are drawn without replacement. (i) What is the probability that both balls are white? (ii) What is the probability that the second ball is red? (iii) What is the probability that the second ball is white? (iv) What is the probability that given the second ball is red, the first ball was also red? (v) What is the probability that given the second ball is red, the first ball was white? Question 6 The problem of colour blindness is 0.02 for a man and 0.001 for a woman. Find the probability that a person picked at random is colour blind if the population is 53% men. Question 7 An insurance company believes that 30% of drivers are careless, and that the probability of a driver having an accident in any one year is 0.4 for a careless driver and 0.2 for a careful driver. Find the probability that a driver will have an accident next year given that she has had an accident this year.

1

(6) An electrical circuit contains 5 components, one of which is faulty. To isolate the fault, the components are tested one by one until the faulty one is found. The random variable X denotes the number of tests requires to locate the fault. The test of the faulty component itself is always included, so that X takes values from 1 to 5 inclusive. Find the expectation and variance of X. The cost C (in suitable units) of locating a fault depends in part on the number of tests required and is given by C = 5 + 2X. Find the expectation and variance of C. One hundred circuits of the above types are to be tested. Find the probability that the average cost will exceed 11.5 units. (7) A coin and a six-faced die are thrown simultaneously. The random variable X is defined as follows: If the coin shows a head, then X is the score on the die, and if the coin shows a tail, then X is twice the score on the die. Find the expected value, , of X and show that P(X < ) = 7/12. Show that Var(X) = 497/48. The experiment is repeated and the sum of the two values obtained for X is denoted by Y. Find P(Y = 4) and E(Y).

R JUGURNAUTH

UOM

ELEC 2001Y Analytical Techniques Probability Problem Sheet 4 Discrete Random Variables(1) An instructor who taught two sections of statistics last term, the first with 20 students and the second with 30, decided to assign a term project. After all projects had been turned in, the instructor randomly ordered them for grading. Consider the first 15 graded projects. (a) What is the probability that at least 10 of these are from the second section? (b) What is the probability that at least 10 of these are from the same section? (c) What are the mean value and standard deviation of the number among these 15 that are from the second section? (d) What are the mean value and standard deviation of the number of projects not among these first 15 that are from the second section? (2) A manufacturer sells a certain article in batches of 5000. By agreement with a customer, the following method of inspection is adopted: a sample of 100 items is drawn at random from each batch and inspected. If the sample contains 4 or fewer defective items, then the batch is accepted by the customer. If more than 4 defective items are found, every item in the batch is inspected. If inspection costs are 75 cents per hundred articles, and the manufacturer normally produces 2% of defective articles, find the average inspection cost per batch. (3) In one part of a country, one person in 80 has blood of Type P. A random sample of 150 blood donors is chosen from that part of the country. Let X represent the number of donors in the sample having blood of Type P. (a) State the distribution of X. Find the parameter of the Poisson distribution which can be used as an approximation. Give a reason why a Poisson distribution is appropriate. (b) Using the Poisson distribution, calculate the probability that in the sample of 150 donors, at least two have blood of Type P. (c) A hospital urgently requires blood of Type P. How large a random sample of donors must be taken in order that the probability of finding at least one donor of Type P must be 0.99 or more. (4) A randomly chosen doctor in general practice sees, on average, one case of a broken nose per year and each case is independent of other similar cases. (a) Regarding a month as a twelfth part of a year, (i) show that the probability that, between them, three such doctors see no cases of a broken nose in a period of one month is 0.779, correct to three significant figures, (ii) find the variance of the number of cases seen by three such doctors in a period of six months (b) Find the probability that, between them, three such doctors see at least three cases in one year (c) Find the probability that, of three such doctors, one sees three cases and the other two see no cases in one year. (5) Before starting to play the game Snake and Ladders each player throws an ordinary unbiased die until a six is obtained. The number of throws before a player starts is the random variable Y, where Y takes the values 1, 2, 3, (a) Name the probability distribution of Y. (b) Find Var(Y). (c) Two people play Snakes and Ladders. Calculate the probability that they will each need at least five throws before starting. (6) During a weekday, heavy lorries pass a point P on a village high street independently and at random times. The mean rate for westward traveling lorries is two in any 30-minute period, and for eastward traveling lorries is three in any 30-minute period. Find the probability (a) that there will be no lorries passing P in a given 10-minute period (b) that at least one lorry from each direction will pass P in a given 10-minute period (c) that there will be exactly four lorries passing P in a given 20-minute period.

R JUGURNAUTH

UOM

(a) Use an unbiased estimate of variance to calculate an approximate 90 % confidence interval for the mean mass (in grams) of all one-rupee coins, giving the end-values of the interval to two decimal places. (b) Estimate the size of a random sample of one-rupee coins that would be required to give a 95 % confidence interval whose width is half that of the interval calculated in (a).

R JUGURNAUTH

UOM

Question 4 (a) Let S be the set of all possible outcomes of an experiment. Assume A1, A2 and A3 are mutually exclusive and exhaustive events within the sample space S. Let D be any other event. Illustrate this scenario with the aid of a Venn diagram, and thus explain in set notation (using and symbols) the meanings of the words (i) mutually exclusive and (ii) exhaustive in the context of the experiment. [3 marks] With the aid of a Venn diagram explain why for any two events A and B, [3 marks] P(A|B) = P(AB)/P(B) and P(B|A) = P(BA)/P(A). (i) If A and B are two events such that P ( A) 0 and P ( B ) 0 , prove that if P( A B) > P( A) then P( B A) > P( B) . (ii) IfA, B

(b)

(c)

[3 marks]

and C are three events such that P (C ) 0 and [2 marks]

P ( B C ) 0 , show that P( A B C ) = P( A B C ) P( B C ) P(C ) .

(d) For A1, A2, A3 and D in part (a), since P( D) = P( D A1 ) + P( D A2 ) + P( D A3 ) , then starting with the result in part (b), prove Bayes Rule:P( A1 | D) = P( A1) P( D | A1) . P( D | A1) P( A1) + P( D | A2 )P( A2 ) + P(D | A3 ) P( A3 )

[6 marks] (e) Let 100 bits/sec be received from transmitter A1 with error rate =1/5; let 200 bits/sec be received from transmitter A2 with error rate =1/10; let 400 bits/sec be received from transmitter A3 with error rate =1/40. Let all three bit streams be randomly multiplexed together, and let one bit be selected at random from the composite signal. Let D represent the event that this bit is in error. By using the results from part (d): (i) (ii) What is the probability (P(D)) that the selected bit is in error? [4 marks] If the selected bit is in error, what is the probability that it is from transmitter A1? [4 marks]

Page 4 of 7

Question 5 (a) Let the voltage V(t) represent some parameter in a communications system. Let it also be a continuous random variable with pdf:

a , for 3 . pV ( ) = > 0, for 3 (i) Sketch pV ( ) , and from this explain why the value of the constant a must equal 1/9; [2 marks]2 Determine the mean, V , the variance, V , and the power ( V 2 ) of V(t); [4 marks]

(ii)

(iii) Calculate the probability that the voltage has a value 1 V (t ) 2 . [2 marks] (b) A shop sells rose plants in boxes, each box containing the same number of plants. The number of plants per box which produce yellow flowers has a binomial distribution with mean 11 and variance 4.95. (i) (ii) Calculate the number of plants per box. [2 marks]

Calculate the probability that a box contains exactly 12 plants which produce yellow flowers. [2 marks]

(iii) Another shop sells rose plants in boxes of 100. The shops advertisement states that the probability of any rose plant producing a pink flower is 0.3. Use a suitable approximation to calculate the probability that a box contains fewer than 35 plants which produce pink flowers. [3 marks] (c) The probability distribution function for a Poisson random variable with parameter is given by

P( X = k ) = e(i)

kk!

k = 0,1,.........

Under what conditions does this probability distribution function arise. [2 marks] For a Poisson random variable X 2 that X = and X =. with parameter show [8 marks]

(ii)

Page 5 of 7

Question 6 (a) A packet contains six biscuits, each of which is individually wrapped. The mass of a biscuit can be taken to be normally distributed with mean 70 g and standard deviation 4 g. The mass of the individual wrapping of a biscuit is normally distributed with mean 10 g and standard deviation 1 g. The mass of the outer packaging is normally distributed with mean 30 g and standard deviation 3 g. Assuming that the masses of the biscuits, wrappings and packaging are independent, calculate the probability that the total mass of a randomly chosen packet and its contents lies between 500 g and 520 g. [6 marks] An experiment carried out to compare the bond strength of modified mortar to that of unmodified mortar resulted in x = 18.12 kgf/cm2 for the modified mortar (sample size n1 = 40) and y = 16.87 kgf/cm2 for the unmodified mortar (sample size n 2 = 32). Assume that the bond strength distributions are both normal. (i) Assuming that standard deviations 1 = 1.6 and 2 = 1.4 for the modified mortar and the unmodified mortar respectively, test at level 0.01 if the bond strength for modified mortar is greater than that for the unmodified mortar. [4 marks]

(b)

(ii) Calculate the probability of a type II error for the test of part (i) when mean difference between the bond strengths of modified mortar and unmodified mortar is actually 1 2 = 1. [4 marks] (iii) How would the analysis and conclusions of part (i) change if the population standard deviations 1 and 2 were unknown, but instead the sample standard deviations s1 = 1.6 and s2 = 1.4 were known? [4 marks] (c) Each helmet in a random sample of 42 helmets was subjected to a certain impact test and 27 showed damage. (i) Calculate a 99 % Confidence Interval of the proportion of all helmets that will show damage when tested in this manner. [4 marks]

(ii) What sample size would be required for the width of a 99 % Confidence Interval to be at most 0.10? [3 marks]

END OF QUESTION PAPER

Page 6 of 7

Page 7 of 7