Combining. random variables

Transcription

1 7 Combiig radom variables If you kow the average height of a brick, the it is fairly easy to guess the average height of two bricks, or the average height of half of a brick. What is less obvious is the variatio of these heights. Eve if we ca predict the mea ad the variace of this radom variable this is ot eough to fid the probability of it takig a particular value. To do this, we also eed to kow the distributio of the radom variable. There are some special cases where it is possible to fid the distributio of the radom variable, but i most cases we meet the eormous sigificace of the ormal distributio; if the sample is large eough, the sample average will (early) always follow a ormal distributio. 7A Addig ad multiplyig all the data by a costat The average height of the studets i a class is.75 m ad their stadard deviatio is 0. m. If they all the stood o their 0.5 m tall chairs the the ew average height would be.5 m, but the rage, ad ay other measure of variability, would ot chage, ad so the stadard deviatio would still be 0. m. If we add a costat to all the variables i a distributio, we add the same costat o to the expectatio, but the variace does ot chage: I this chapter you will lear: how multiplyig all of your data by a costat or addig a costat chages the mea ad the variace how addig or multiplyig together two idepedet radom variables chages the mea ad the variace how we ca apply these ideas to makig predictios about the average or the sum of a sample about the distributio of liear combiatios of ormal variables about the distributio of the sum or average of lots of observatios from ay distributio. E ( X c) E( X) c ( c) Var( X) Var If, istead, each studet were give a magical growig potio that doubled their heights, the ew average height would be 3.5 m, ad i this case the rage (ad ay similar measure of variability) would also double, so the ew stadard deviatio would be 0. m. This meas that their variace would chage from 0.0 m to 0.04 m. Cambridge Mathematics for the IB Diploma Higher Level Cambridge Uiversity Press, 0 Optio 7: 7 Combiig radom variables

2 exam hit It is importat to kow that this oly works for the structure ax + c which is called a liear fuctio. So, for example, E( X ) caot be simplified to E( X) ad E( X ) is ot equivalet to. E X If we multiply a radom variable by a costat, we multiply the expectatio by the costat ad multiply the variace by the costat squared: E( a ) aee( X) Var a a Var X These ideas ca be combied together: KEY POINT 7. E ( ax c) ae( X ) c ( a c) a Var( X) Var Worked example 7. A piece of pipe with average legth 80 cm ad stadard deviatio cm is cut from a 00 cm legth of water pipe. The leftover piece is used as a short pipe. Fid the mea ad stadard deviatio of the legth of the short pipe. Defie your variables L = crv legth of log pipe S = crv legth of short pipe Write a equatio to coect the variables S = 00 L Apply expectatio algebra E(S) = E(00 L) = 00 E(L) = = 0 So the mea of S is 0 cm Var(S) = Var (00 L) = ( ) Var(L) = Var(L) = 4 cm So the stadard deviatio of S is also cm. exam hit Eve if the coefficiets are egative, you will always get a positive variace (sice square umbers are always positive). If you fid you have a egative variace, somethig has goe wrog! The result regardig E(aX + b) stated i Key poit 7. represets a more geeral result about the expectatio of a fuctio of a radom variable: Cambridge Mathematics for the IB Diploma Higher Level Cambridge Uiversity Press, 0 Optio 7: 7 Combiig radom variables

3 KEY POINT 7. For a discrete radom variable: = g E g X) i p i For a cotiuous radom variable with probability desity fuctio f x) x : E( g ) g x f dx You have used this before whe fidig E( X ). See Key poit 3.4. Worked example 7. The cotiuous radom variable X has probability desity e x for 0 < x < l. The radom X variable Y is related to X by the fuctio Y = e. Fid E( Y ). Use the formula for the expectatio of a fuctio of a variable l ) 0 E e x e edx Use the laws of expoets l 0 = e dx l = [ ] 0 l = e ( ) = + = Exercise 7A. If X = 4 fid: (a) (i) E3X (ii) E6X (b) (i) E X (ii) E 3 X 4 (c) (i) E( X ) (ii) E( 4X) (d) (i) E( X + 5 ) (ii) E( X 3) (e) (i) E5 X (ii) E3X +. If Var X = 6 fid: (a) (i) Var( 3X) (ii) Var( 6X) X (b) (i) Var (ii) Var 3 X 4 (c) (i) Var( X ) (ii) Var( 4X) (d) (i) Var( X + 5 ) (ii) Var( X 3) (e) (i) Var 5 X (ii) Var 3X + Cambridge Mathematics for the IB Diploma Higher Level Cambridge Uiversity Press, 0 Optio 7: 7 Combiig radom variables 3

4 3. The probability desity fuctio of the cotiuous radom variable Z is kz for < z 3. (a) Fid the value of k. (b) Fid E(Z). (c) Fid E( 6Z + 5). (d) Fid the exact value of E + Z. [0 marks] See Sectio D for a remider of what is meat by idepedet. 7B Addig idepedet radom variables A teis racquet is made by addig together two compoets; the hadle ad the head. If both compoets have their ow distributio of legth ad they are combied together radomly the we have formed a ew radom variable: the legth of the racquet. It is ot surprisig that the average legth of the whole racquet is the sum of the average legths of the parts, but with a little thought we ca reaso that the stadard deviatio will be less tha the sum of the stadard deviatio of the parts. To get either extremely log or extremely short teis racquets we must have extremes i the same directio for both the hadle ad the head. This is ot very likely. It is more likely that either both are close to average or a extreme value is paired with a average value or a extreme value i oe directio is balaced by aother. See Fill-i proof 8 Expectio of a sum of idepedet variables o the CD-ROM for the proof of these results. KEY POINT 7.3 Liear Combiatios E( ax ax ) ae E X ) ± ae X ) Var a a a Var X + a Var ( ) a X The result for variace is oly true if X ad Y are idepedet. There is a similar result for the product of two idepedet radom variables: KEY POINT 7.4 If X ad Y are idepedet radom variables the: E E( X E Y) We could write the whole of statistics oly usig stadard deviatio, without referrig to variace at all, where σ( X + by + ) σ ( X) b σ ( Y ). However, as you ca see, the cocept of stadard deviatio squared occurs very aturally. Is this a sufficiet justificatio for the cocept of variace? 4 Cambridge Mathematics for the IB Diploma Higher Level Cambridge Uiversity Press, 0 Optio 7: 7 Combiig radom variables

5 It is ot immediately obvious that if Var(X + Y) = Var(X) + Var(Y) the the stadard deviatio of (X + Y) will always be less tha the stadard deviatio of X plus the stadard deviatio of Y. This is a example of oe of may iterestig iequalities i statistics. Aother is that E( ) E ( X ) which esures that variace is always positive. If you are iterested i provig these types of iequalities you might like to look at the Cauchy- Schwarz iequality. exam hit Notice i particular that, if X ad Y are idepedet: (Y ) Var Var X Var Y = Var + V ar ( Y ) The result exteds to more tha two variables. Worked example 7.3 The mea thickess of the base of a burger bu is.4 cm with variace 0.0 cm. The mea thickess of a burger is 3.0 cm with variace 0.4 cm. The mea thickess of the top of the burger bu is. cm with variace 0. cm. Fid the mea ad stadard deviatio of the total height of the whole burger ad bu, assumig that the thickess of each part is idepedet. Defie your variables Write a equatio to coect the variables X = crv Thickess of base Y = crv Thickess of burger Z = crv Thickess of top T = crv Total thickess T = X + Y + Z E( + ) Apply expectatio algebra E E X Y Z = E + E(Y + E( Z ) = 6.6 cm So the mea of T is 6.6 cm Var Var( X + Y Z) = Var + V ar( (Y) + Var( Z ) = 0.36 cm So stadard deviatio of T is 0.6 cm X ad Y have to be idepedet (see Key poit 7.3) but this does ot mea that they have to be draw from differet populatios. They could be two differet observatios of the same populatio, for example the heights of two differet people added together. This is a differet variable from the height of Cambridge Mathematics for the IB Diploma Higher Level Cambridge Uiversity Press, 0 Optio 7: 7 Combiig radom variables 5

6 oe perso doubled. We will use a subscript to emphasise whe there are repeated observatios from the same populatio: X + X meas addig together two differet observatios of X X meas observig X oce ad doublig the result. The expectatio of both of these combiatios is the same, E(X), but the variace is differet. From Key poit 7.3: ( ) ( ) Var Var X Var X From Key poit 7.: = Var( X) Var Var ( X) So the variability of a sigle observatio doubled is greater tha the variability of two idepedet observatios added together. This is cosistet with the earlier argumet about the possibility of idepedet observatios cacellig out extreme values. Worked example 7.4 I a office, the mea mass of the me is 84 kg ad stadard deviatio is kg. The mea mass of wome i the office is 64 kg ad the stadard deviatio is 6 kg. The wome thik that if four of them are picked at radom their total mass will be less tha three times the mass of a radomly selected ma. Fid the mea ad stadard deviatio of the differece betwee the sums of four wome s masses ad three times the mass of a ma, assumig that all these people are chose idepedetly. Defie your variables X = crv Mass of a ma Y = crv Mass of a woma D = crv Differece betwee the mass of 4 wome ad 3 lots of ma Write a equatio to coect the variables D Y + Y Y + Y 3X Y 3 4 = ) Apply expectatio algebra ED EY EY EY 3 ) + EY ( 3( E X) = 4 kg ( Y ) + ( ) ( 3 ) + Var D Var Y Var Var Y Var( 4 ) ( 3 ) Var( X) = 33 kg So the stadard deviatio of D is 35. kg Fidig the mea ad variace of D is ot very useful uless you also kow the distributio of D. I Sectios 7D ad 7E you will see that this ca be doe i certai circumstaces. We ca the go o to calculate probabilities of differet values of D. 6 Cambridge Mathematics for the IB Diploma Higher Level Cambridge Uiversity Press, 0 Optio 7: 7 Combiig radom variables

7 Exercise 7B =,. Let X ad Y be two idepedet variables with E X Var( X) =, E( Y ) = 4 ad Var( Y ) = 4. Fid the expectatio ad variace of: (a) (i) X Y (ii) X Y (b) (i) 3X Y (ii) X 4Y (c) (i) X 3 Y + X Y (ii) 5 3 Deote by X i, Y i idepedet observatios of X ad Y. (d) (i) X X + X3 (ii) Y Y (e) (i) X X Y (ii) 3X ( Y + Y Y 3). If X is the radom variable mass of a gerbil explai the differece betwee X ad X X. 3. Let X ad Y be two idepedet variables with E( X ) = 4, Var( X) =, E( Y ) = ad Var( Y ) = 6. Fid: (a) E3X (b) Var 3X ( X Y + ) (c) E3 ( X Y + ) (d) Var 3 [6 marks] 4. The average mass of a ma i a office is 85 kg with stadard deviatio kg. The average mass of a woma i the office is 68 kg with stadard deviatio 8 kg. The empty lift has a mass of 500 kg. What is the expectatio ad stadard deviatio of the total mass of the lift whe 3 wome ad 4 me are iside? [6 marks] 5. A weighted die has mea outcome 4 with stadard deviatio. Bria rolls the die oce ad doubles the outcome. Camilla rolls the die twice ad adds her results together. What is the expected mea ad stadard deviatio of the differece betwee their scores? [7 marks] 6. Exam scores at a large school have mea 6 ad stadard deviatio 8. Two studets are selected at radom. Fid the expected mea ad stadard deviatio of the differece betwee their exam scores. [6 marks] 7. Adria cycles to school with a mea time of 0 miutes ad a stadard deviatio of 5 miutes. Pamela walks to school with a mea time of 30 miutes ad a stadard deviatio of miutes. They each calculate the total time it takes them to get to school over a five-day week. What is the expected mea ad stadard deviatio of the differece i the total weekly jourey times, assumig jourey times are idepedet? [7 marks] Cambridge Mathematics for the IB Diploma Higher Level Cambridge Uiversity Press, 0 Optio 7: 7 Combiig radom variables 7

8 8. I this questio the discrete radom variable X has the followig probability distributio: x 3 4 P(X = x) k (a) Fid the value of k. (b) Fid the expectatio ad variace of X. (c) The radom variable Y is give by Y 6 X. Fid the expectatio ad the variace of Y. (d) Fid E(XY) ad explai why the formula E E( X E Y) is ot applicable to these two variables. (e) The discrete radom variable Z has the followig distributio, idepedet of X: z P(Z = z) p p If E( XZ ) = 35 8 fid the value of p. [4 marks] 7C Expectatio ad variace of the sample mea ad sample sum Whe calculatig the mea of a sample of size of the variable X we have to add up idepedet observatios of X the divide by. We give this sample mea the symbol X ad it is itself a radom variable (as it might chage each time it is observed). X X X + X + + = = X X X This is a liear combiatio of idepedet observatios of X, so we ca apply the rules of the previous sectio to get the followig very importat results: KEY POINT 7.5 E E( X) Var X Var( X ) = The first of these results seems very obvious; the average of a sample is, o average, the average of the origial variable, but you will see i chapter 30 that this is ot the case for all sample statistics. 8 Cambridge Mathematics for the IB Diploma Higher Level Cambridge Uiversity Press, 0 Optio 7: 7 Combiig radom variables

9 The secod result demostrates why meas are so importat; their stadard deviatio (which ca be thought of as a measure of the error caused by radomess) is smaller tha the stadard deviatio of a sigle observatio. This proves mathematically what you probably already kew istictively, that fidig a average of several results produces a more reliable outcome tha just lookig at oe result. The result actually goes further tha that; it cotais what ecoomists call The law of dimiishig returs. The stadard deviatio of the mea is proportioal to, so goig from a sample of to a sample of 0 has a much bigger impact tha goig from a sample of 0 to a sample of 0. Worked example 7.5 Prove that if X is the average of idepedet observatios of X the Var ( X ) = Var X. Write X i terms of X i X X + + X X = = ( + + ) X X + X Apply expectatio algebra Var Var ( X + + X X ) = Var X X X = ( ) ) ( times Sice X X, are all observatios of X = Var + Var + + Var = ( ) = Var( X ) We ca apply similar ideas to the sample sum. KEY POINT 7.6 For the sample sum: E i E( X ) ad Var i Var( X) See Key poit 7. for a remider about sigma otatio. Cambridge Mathematics for the IB Diploma Higher Level Cambridge Uiversity Press, 0 Optio 7: 7 Combiig radom variables 9

10 Exercise 7C The ormal distributio was studied i Sectio 4C of the course book. The biomial ad Poisso distributio were studied i chapter 3.. A sample is obtaied from idepedet observatios of a radom variable X. Fid the expected value ad the variace of the sample mea i the followig situatios: (a) (i) ( X 5, V X X) =., = 7 (ii) E( X 6, V X) X =. 5, = (b) (i) E( X. 7, Var X) = 08., = 0 (ii) E( X 5., V X) = 0. 7, = 5 (c) (i) X N, ), 0 (ii) X N,., 4 (d) (i) X N,., 7 (ii) X N,., 5 (e) (i) X B,., 0 (ii) X B(,. ), = 8 (f) (i) X P, = 0 (ii) X P, 5 =. Fid the expected value ad the variace of the total of the samples from the previous questio. 3. Eggs are packed i boxes of. The mass of the box is 50 g. The mass of oe egg has mea.4 g ad stadard deviatio. g. Fid the mea ad the stadard deviatio of the mass of a box of eggs. [4 marks] 4. A machie produces chocolate bars so that the mea mass of a bar is 0 g ad the stadard deviatio is 8.6 g. As a part of the quality cotrol process, a sample of 0 chocolate bars is take ad the mea mass is calculated. Fid the expectatio ad variace of the sample mea of these 0 chocolate bars. [5 marks] 5. Prove that Var i Var( X ). [4 marks] 6. The stadard deviatio of the mea mass of a sample of aubergies is 0 g smaller tha the stadard deviatio i the mass of a sigle aubergie. Fid the stadard deviatio of the mass of a aubergie. [5 marks] 7. A radom variable X takes values 0 ad with probability 4 ad 3 4, respectively. (a) Calculate E( X ) ad Var( X ). 0 Cambridge Mathematics for the IB Diploma Higher Level Cambridge Uiversity Press, 0 Optio 7: 7 Combiig radom variables

11 A sample of three observatios of X is take. (b) List all possible samples of size 3 ad calculate the mea of each. (c) Hece complete the probability distributio table for the sample mea, X. x P( X x) 64 (d) Show that E E( X ) ad Var ( X ) = Var( X). [4 marks] 3 8. A laptop maufacturer believes that the battery life of the computers follows a ormal distributio with mea 4.8 hours ad variace.7 hours. They wish to take a sample to estimate the mea battery life. If the stadard deviatio of the sample mea is to be less tha 0.3 hours, what is the miimum sample size eeded? [5 marks] 9. Whe the sample size is icreased by 80, the stadard deviatio of the sample mea decreases to a third of its origial size. Fid the origial sample size. [4 marks] 7D Liear combiatios of ormal variables Although the proof is beyod the scope of this course, it turs out that ay liear combiatio of ormal variables will also follow a ormal distributio. We ca use the methods of Sectio C to fid out the parameters of this distributio. KEY POINT 7.7 If X ad Y are radom variables followig a ormal distributio ad Z ax by c the Z also follows a ormal distributio. Worked example 7.6 If X ~ N (, ), Y ~ N (, ) ad Z = X + Y + 3fid P( Z ). Use expectatio algebra E E( X ) + E + 3= 7 Var Z Var X Var Y 8 + = 7 State distributio of Z Z ~ ( 7, ) P Z > ( from GDC) = Cambridge Mathematics for the IB Diploma Higher Level Cambridge Uiversity Press, 0 Optio 7: 7 Combiig radom variables

12 Worked example 7.7 If X ~ N ( 5, ) ad four idepedet observatios of X are made fid P( X ). Express X i terms of observatios of X Use expectatio algebra E Var X X + X X X = 4 E X = ( X ) 4 = 5 ( X ) = Var = 36 X 3 4 ( X ) 4 State distributio of X X ~ N ( 5, ) P X < = ( from GDC) I Sectio 3D of the coursebook you met the idea that the Poisso distributio was scaleable. We ca ow iterpret this as meaig that the sum of two Poisso variables is also Poisso. However, this oly applies to sums of Poisso distributios, ot differeces or multiples or liear combiatios. Exercise 7D exam hit Make sure you do ot cofuse the stadard deviatio ad the variace!. If X ~ N (, ) ad Y ~ N ( 8, ), fid: (a) (i) P( X Y > ) (ii) P( X Y < 4) (b) (i) P3X Y 50) (ii) P( X 3 Y > ) (c) (i) P( X Y ) (ii) PX 3 Y) (d) (i) P( X Y ) (ii) P( 3X 5Y) (e) (i) P( X X > X ) X X 3 X (ii) P( X Y + Y X + ) Y X (f) (i) P( X > ) where X is the average of observatios of X (ii) P( Y < 6 ) where Y is the average of 9 observatios of Y. A airlie has foud that the mass of their passegers follows a ormal distributio with mea 8. kg ad variace 0.7 kg. The mass of their had luggage follows a ormal distributio with mea 9. kg ad variace 5.6 kg. (a) State the distributio of the total mass of a passeger ad their had luggage ad fid ay ecessary parameters. (b) What is the probability that the total mass of a passeger ad their luggage exceeds 00 kg? [5 marks] Cambridge Mathematics for the IB Diploma Higher Level Cambridge Uiversity Press, 0 Optio 7: 7 Combiig radom variables

13 3. Evidece suggests that the times Aaro takes to ru 00 m are ormally distributed with mea 3. s ad stadard deviatio 0.4 s. The times Bashir takes to ru 00 m are ormally distributed with mea.8 s ad stadard deviatio 0.6 s. (a) Fid the mea ad stadard deviatio of the differece (Aaro Bashir) betwee Aaro s ad Bashir s times. (b) Fid the probability that Aaro fiishes a 00 m race before Bashir. (c) What is the probability that Bashir beats Aaro by more tha secod? [7 marks] 4. A machie produces metal rods so that their legth follows a ormal distributio with mea 65 cm ad variace 0.03 cm. The rods are checked i batches of six, ad a batch is rejected if the mea legth is less tha 64.8 cm or more tha 65.3 cm. (a) Fid the mea ad the variace of the mea of a radom sample of six rods. (b) Hece fid the probability that a batch is rejected. [5 marks] 5. The legths of pipes produced by a machie is ormally distributed with mea 40 cm ad stadard deviatio 3 cm. (a) What is the probability that a radomly chose pipe has a legth of 4 cm or more? (b) What is the probability that the average legth of a radomly chose set of 0 pipes of this type is 4 cm or more? [6 marks] 6. The masses, X kg, of male birds of a certai species are ormally distributed with mea 4.6 kg ad stadard deviatio 0.5 kg. The masses, Y kg, of female birds of this species are ormally distributed with mea.5 kg ad stadard deviatio 0. kg. (a) Fid the mea ad variace of Y X. (b) Fid the probability that the mass of a radomly chose male bird is more tha twice the mass of a radomly chose female bird. (c) Fid the probability that the total mass of three male birds ad 4 female birds (chose idepedetly) exceeds 5 kg. [ marks] 7. A shop sells apples ad pears. The masses, i grams, of the apples may be assumed to have a N ( 80, ) distributio ad the masses of the pears, i grams, may be assumed to have a 00, 0 distributio. N (a) Fid the probability that the mass of a radomly chose apple is more tha double the mass of a radomly chose pear. (b) A shopper buys apples ad a pear. Fid the probability that the total mass is greater tha 500 g. [0 marks] Cambridge Mathematics for the IB Diploma Higher Level Cambridge Uiversity Press, 0 Optio 7: 7 Combiig radom variables 3

14 8. The legth of a corsake is ormally distributed with mea. m. The probability that a radomly selected sample of 5 corsakes havig a average of above.4 m is 5%. Fid the stadard deviatio of the legth of a corsake. [6 marks] 9. (a) I a test, boys have scores which follow the distributio N(50, 5). Girls scores follow N(60, 6). What is the probability that a radomly chose boy ad a radomly chose girl differ i score by less tha 5? (b) What is the probability that a radomly chose boy scores less tha three quarters of the mark of a radomly chose girl? [0 marks] 0. The daily raifall i Algebraville follows a ormal distributio with mea μ mm ad stadard deviatio σ mm. The raifall each day is idepedet of the raifall o other days. O a radomly chose day, there is a probability of 0. that the raifall is greater tha 8 mm. I a radomly chose 7-day week, there is a probability of 0.05 that the mea daily raifall is less tha 7 mm. Fid the value of μ ad of σ. [7 marks]. Au uses public trasport to go to school each morig. The time she waits each morig for the trasport is ormally distributed with a mea of miutes ad a stadard deviatio of 4 miutes. (a) O a specific morig, what is the probability that Au waits more tha 0 miutes? (b) Durig a particular week (Moday to Friday), what is the probability that (i) (ii) her total morig waitig time does ot exceed 70 miutes? she waits less tha 0 miutes o exactly morigs of the week? (iii) her average morig waitig time is more tha 0 miutes? (c) Give that the total morig waitig time for the first four days is 50 miutes, fid the probability that the average for the week is over miutes. (d) Give that Au s average morig waitig time i a week is over 4 miutes, fid the probability that it is less tha 5 miutes. [0 marks] 4 Cambridge Mathematics for the IB Diploma Higher Level Cambridge Uiversity Press, 0 Optio 7: 7 Combiig radom variables

15 7E The distributio of sums ad averages of samples I this sectio we shall look at how to fid the distributio of the sample mea or the sample total, eve if we do ot kow the origial distributio. The graph alogside shows 000 observatios of the roll of a fair die. It seems to follow a uiform distributio quite well, as we would expect. However, if we look at the sum of dice 000 times the distributio looks quite differet. f f x f The sum of 0 dice is startig to form a more familiar shape. The sum seems to form a ormal distributio. This is more tha a coicidece. If we sum eough idepedet observatios of ay radom variable, the result will follow a ormal distributio. This result is called the Cetral Limit Theorem or CLT. We geerally take 30 to be a sufficietly large sample size to apply the CLT. As we saw i Sectio 7D, if a variable is ormally distributed the a multiple of that variable will also be ormally distributed. Sice X = Xi it follows that the mea of a sufficietly large sample is also ormally distributed. Usig Key poit 7.5 where Var( X) E E( X ) ad Var( X ) =, we ca predict which ormal distributio is beig followed: KEY POINT 7.8 Cetral Limit Theorem For ay distributio if E( X) =μ, Var( X ) =σ ad 30, the the approximate distributios of the sum ad the mea are give by: X i ~ N ( ) σ X ~ N, x x There There are may are may other other distributios distributios which which have have a a similar similar shape, shape, such as such the as the Cauchy distributio. Cauchy distributio. To show that To show that these sums these form a sums ormal form a ormal distributio distributio we eed to we use eed momet to use geeratig momet fuctios, geeratig which are fuctios, well beyod which this course. are well beyod this course Cambridge Mathematics for the IB Diploma Higher Level Cambridge Uiversity Press, 0 Optio 7: 7 Combiig radom variables 5

16 Worked example 7.8 Esme eats a average of 900 kcal each day with a stadard deviatio of 400 kcal. What is the probability that i a 3-day moth she eats more tha 000 kcal per day o average? Check coditios for CLT are met State distributio of the mea Calculate the probability Sice we are fi dig a average over 3 days we ca use the CLT. X ~ N 900, X > = ( 3SF from GDC) P Exercise 7E. The radom variable X has mea 80 ad stadard deviatio 0. State where possible the approximate distributio of: (a) (i) X if the sample has size. (ii) X if the sample has size 3. (b) (i) X if the average is take from 00 observatios. (ii) X if the average is take from 400 observatios. (c) (i) 50 X i (ii) 50 X i. The radom variable Y has mea 00 ad stadard deviatio 5. A sample of size is foud. Fid, where possible, the probability that: (a) (i) P( Y < ) if = 00 (ii) P( Y < ) if = 00 (b) (i) P( Y < ) if = (ii) P( Y < ) if = 3 (c) (i) P( Y 95 > 0) if = 00 (ii) P( Y 0 > 3 ) if = (d) (i) P Y > 0, 500 (ii) P i Y 9, 500 i 3. Radom variable X has mea ad stadard deviatio 3.5. A sample of 40 idepedet observatios of X is take. Use the Cetral Limit Theorem to calculate the probability that the mea of the sample is betwee 3 ad 4. [5 marks] 4. The weight of a pomegraate, i grams, has mea 45 ad variace 96. A crate is filled with 70 pomegraates. What is the probability that the total weight of the pomegraates i the crate is less tha 0 kg? [5 marks] 5. Give that X ~ 6, fid the probability that the mea of 35 idepedet observatios of X is greater tha 7. [6 marks] 6 Cambridge Mathematics for the IB Diploma Higher Level Cambridge Uiversity Press, 0 Optio 7: 7 Combiig radom variables

17 6. The average mass of a sheet of A4 paper is 5 g ad the stadard deviatio of the masses is 0.08 g. (a) Fid the mea ad stadard deviatio of the mass of a ream of 500 sheets of A4 paper. (b) Fid the probability that the mass of a ream of 500 sheets is withi 5 g of the expected mass. (c) Explai how you have used the Cetral Limit Theorem i your aswer. [7 marks] 7. The times Markus takes to aswer a multiple choice questio are ormally distributed with mea.5 miutes ad stadard deviatio 0.6 miutes. He has oe hour to complete a test cosistig of 35 questios. (a) Assumig the questios are idepedet, fid the probability that Markus does ot complete the test i time. (b) Explai why you did ot eed to use the Cetral Limit Theorem i your aswer to part (a). [6 marks] 8. A radom variable has mea 5 ad stadard deviatio 4. A large umber of idepedet observatios of the radom variable is take. Fid the miimum sample size so that the probability that the sample mea is more tha 6 is less tha [8 marks] Summary Whe addig ad multiplyig all the data by a costat: the expectatio of variables geerally behaves as you would expect: E( ax c) ae( X ) c E( ax ax ) ae E X ) ± ae X ) the variace is more subtle: Var( a c) a Var( X) Var( a a ) a Var ( X a ) + Var( X ) whe X ad X are idepedet. A more geeral result about the expectatio of a fuctio of a discrete radom variable is: E g X) g p. For a cotiuous radom variable with probability desity fuctio = ( x i ) i i x ) ) = f x :E g X g x f x) d x. For the sum of idepedet radom variables: E(a X ± a X ) = a E(X ) ± a E(X ) Var(a X ± a X ) = a Var(X ) ± a Var(X ), ote that Var(X Y) = Var(X) + Var(Y). For the product of two idepedet variables: E(XY) =E(X)E(Y). For a sample of observatios of a radom variable X, the sample mea X is a radom variable with mea E E( X ) ad variace Var ( X ) = Var( X). Cambridge Mathematics for the IB Diploma Higher Level Cambridge Uiversity Press, 0 Optio 7: 7 Combiig radom variables 7

18 For the sample sum E X i E( X ) ad Var i Var( X ). Whe we combie differet variables we do ot ormally kow the resultig distributio. However there are two importat exceptios:. A liear combiatio of ormal variables also follows a ormal distributio. If X ad Y are radom variables followig a ormal distributio ad Z ax by c the Z also follows a ormal distributio.. The sum or mea of a large sample of observatios of a variable follows a ormal distributio, irrespective of the origial distributio this is called the Cetral Limit Theorem. For ay distributio if E( X) = μ, Var( X ) =σ ad 30 the the approximate distributios are give by: X ~ N ( ) σ X ~ N, 8 Cambridge Mathematics for the IB Diploma Higher Level Cambridge Uiversity Press, 0 Optio 7: 7 Combiig radom variables

19 Mixed examiatio practice 7 This chapter does ot usually have its ow examiatio questios, so the examples below are parts of loger examiatio questios.. X is a radom variable with mea μ ad variace σ. Y is a radom variable with mea m ad variace s. Fid i terms of μ, σ, m ad s: (a) E( X Y) (b) Var( X Y) (c) Var( 4X) (d) Var( X X + X3 X4) where X i is the ith observatio of X. [4 marks]. The heights of trees i a forest have mea 6 m ad variace 60 m. A sample of 35 trees is measured. (a) Fid the mea ad variace of the average height of the trees i the sample. (b) Use the Cetral Limit Theorem to fid the probability that the average height of the trees i the sample is less tha m. [5 marks] 3. The umber of cars arrivig at a car park i a five miute iterval follows a Poisso distributio with mea 7, ad the umber of motorbikes follows Poisso distributio with mea. Fid the probability that exactly 0 vehicles arrive at the car park i a particular five miute iterval. [4 marks] 4. The umber of aoucemets posted by a head teacher i a day follows a ormal distributio with mea 4 ad stadard deviatio. Fid the mea ad stadard deviatio of the total umber of aoucemets she posts i a five-day week. [3 marks] 5. The masses of me i a factory are kow to be ormally distributed with mea 80 kg ad stadard deviatio 6 kg. There is a elevator with a maximum recommeded load of 600 kg. With 7 me i the elevator, calculate the probability that their combied weight exceeds the maximum recommeded load. [5 marks] 6. Davia makes bracelets usig purple ad yellow beads. Each bracelet cosists of seve radomly selected purple beads ad four radomly selected yellow beads. The diameters of the beads are ormally distributed with stadard deviatio 0.4 cm. The average diameter of a purple bead is.5 cm ad the average diameter of a yellow bead is. cm. Fid the probability that the legth of the bracelet is less tha 8 cm. [7 marks] Cambridge Mathematics for the IB Diploma Higher Level Cambridge Uiversity Press, 0 Optio 7: 7 Combiig radom variables 9

20 7. The masses of the parets at a primary school are ormally distributed with mea 78 kg ad variace 30 kg, ad the masses of the childre are ormally distributed with mea 33 kg ad variace 6 kg. Let the radom variable P represet the combied mass of two radomly chose parets ad the radom variable C the combied mass of four radomly chose childre. (a) Fid the mea ad variace of C P. (b) Fid the probability that four childre have a mass of more tha two parets. [6 marks] 8. X is a radom variable with mea μ ad variace σ. Prove that the expectatio of the mea of three observatios of X is μ but the stadard deviatio of this mea is σ 3. [7 marks] 9. A aimal scietist is ivestigatig the legths of a particular type of fish. It is kow that the legths have stadard deviatio 4.6 cm. She wishes to take a sample to estimate the mea legth. She requires that the stadard deviatio of the sample mea is smaller tha, ad that the stadard deviatio of the total legth of the sample is less tha. What is the smallest sample size she could take? [6 marks] 0. The marks i a Mathematics test are kow to follow a ormal distributio with mea 63 ad variace 64. The marks i a Eglish test follow a ormal distributio with mea 6 ad variace 7. (a) Fid the probability that a radomly chose mark i Eglish is higher tha a radomly chose Mathematics mark. (b) Fid the probability that the mea of Eglish marks is higher tha the mea of Mathematics marks. [9 marks]. The masses of loaves of bread have mea 80 g ad stadard deviatio σ. The probability that a box cotaiig 40 loaves of bread has mass uder 3 kg is Fid the value of σ. [7 marks] 0 Cambridge Mathematics for the IB Diploma Higher Level Cambridge Uiversity Press, 0 Optio 7: 7 Combiig radom variables