Fial Review for MATH 50 Calculatio 5 Give a fairly simple probability mass fuctio or probability desity fuctio of a radom variable, you should be able to compute the expected value ad variace of the variable For example: a Let 0 < p < Let X be discrete with mass poits k,, ad probability mass fuctio P X k p p k I this case, we say that X has a geometric distributio EX k kp X k EX k k P X k p p p p p k kp pk p p p k k p p k p+ p p b a p p, so varx if a < x < b b Let X be cotiuous with probability desity fuctio fx I this case, we say that X has a uiform distributio EX b a x b a dx b a b a a+b EX b a x b a dx b a b a a +ab+b, so varx EX EX b a x if 0 < x < c Let X be cotiuous with probability desity fuctio fx EX 0 x x dx 5 x 0 5 EX 0 x x dx x6 0, so varx 5 75 Example Problems 6 I have 6 cois i my pocket: are fair, oe is double-headed, oe is double-tailed, ad oe lads heads 75% of the time I flip the coi three times ad get heads each time What is the probability that I am flippig the double-headed coi? Let H be the evet that the coi is double-headed, H that the coi is fair, H that the coi is double-tailed, ad H that the coi lads heads 75% of the time The P H 6, P H 6, P H 6, ad P H 6 Let E be the evet that the coi lads heads i all three flips Usig Bayes Theorem, we calculate: P H E P E H P H P E H P H + P E H P H + P E H P H + P E H P H 6 6 + 8 6 + 0 6 + 075 6 0557
7 A poit Q is chose at radom iside a sphere with radius r What are the expected value ad the stadard deviatio of the distace from the ceter of the sphere to the poit Q? We first calculate the cumulative distributio fuctio for X the distace of Q from the ceter of the sphere Note that for 0 x r, volume of sphere of radius x P X x volume of sphere of radius r πx x πr r So, the probability desity fuctio of X is fx d fx x dx r if 0 < x < r The, EX r 0 xfxdx r 0 so that varx 5 r r 80 r ad σx 096r [P X x] x r x x r dx r r r ad EX r 0 x x r for 0 < x < r, ad dx r5 5r 5 r, 8 Two gamblers, A ad B, play the followig game: each takes a tur drawig a card off the top of a well-shuffled deck util all 5 cards have bee draw Every time a player draws a diamod, the other player gives the drawig player $ Let X be the umber of dollars that A has wo or lost after all 5 cards have bee draw Fid the expected value ad stadard deviatio of X You may leave your aswer usimplified, if you wish Note that we do t care about ay of the cards except the diamods, so we ca focus oly o how may diamods each gambler eds up with at the ed of the game Sice each gambler draws 6 cards, each has a equal probability of gettig each of the diamods Let Ω be the set of -tuples of A ad B that represets which diamods each gambler receives, so that for example A, A, A, B, A, B, B, B, B, B, B, B, A meas that A got the ace, two, three, five, ad kig of diamods ad B got the others Note Ω Let Y be the umber of diamods that A gets The k P Y k for k 0,,,, If Y k, the A will get k dollars from B ad give k dollars to B, so that X k k k Y The k EX EY k P Y k k ad so varx k0 k k0 k0 EX EY k k, k k0 k0 computer, these simplify to EX 0 ad σx 5 k ad σx varx Usig a Fu Fact: Because P Y k is a probability mass fuctio, it must be that k0 P Y k, which implies that k0 More geerally, k0 k k k
9 A bet coi comes up heads with probability p, where 0 < p < a You flip the coi util heads comes up How may times do you expect to flip the coi? Let X be the umber of times you flip the coi If X k, the the first k flips must be tails ad the kth flip must be heads, so P X k p p k This is the geometric distributio, so as we calculated i 5a, we see that EX p b You flip the coi util heads comes up r times How may times do you expect to flip the coi? Let Y be the umber of times you flip the coi to get r heads We say that Y has a egative biomial distributio with parameters r ad p Note that Y X + + X r where X i is the umber of times you flip the coi to go from i heads to i heads We calculated i a that EX i p, so EY EX + + EX r r p 0 Suppose a bridge player s had of cards cotais a ace What is the probability that the player has oly oe ace? Let X be the umber of aces i the players had The probability that the player has oly oe ace give that the had cotais at least oe ace is P X X P X P X P X P X0 If X, the the had cotais oe of the four aces ad the remaiig cards come from 5 the 8 o-aces, so that P X 8 5 the 8 o-aces, so P X 0 8 If X 0, the all cards i the had come from The P X X 06 Let k be a costat ad cosider the fuctio kx if 0 < x < fx a For what values of k is f a probability desity fuctio? Note fxdx 0 kx dx 8k This is a probability desity if ad oly if this itegral equals, so f is a probability desity fuctio if ad oly if k 8 b For the values of k you foud above, suppose X is a radom variable with probability desity fuctio f Fid the expected value ad stadard deviatio of X We compute EX 8 x dx ad EX 8 x dx 96 0, so that varx 96 0 0 x 0, ad σx varx 09 0 x Suppose that U is a umber chose at radom betwee 0 ad ad that X U What is the expected value ad stadard deviatio of X? First, fid the cumulative distributio fuctio of X We compute: P X x P U x P U x x for 0 x x if 0 < x < Takig the derivative, the probability desity fuctio of X is fx So, EX 0 x x dx, EX 0 x x dx 5, so σx 5 096
A had of cards is dealt from a well-shuffled deck How may four-of-a-kids do you expect the had to cotai? How may four-of-a-kids do you expect the 9 cards remaiig i the deck to cotai? Let X be the umber of four-of-a-kids i the card had The X X + + X where X i equals if the had cotais the four-of-a-kid of rak i ad equals We compute EX i P X i 8 9 5 sice the had cotais the four cards of rak i ad the other 9 cards i the had ca be ay of the other cards i the deck The EX 8 5 00 Now, let X be the umber of four-of-a-kids i the cards remaiig i the deck Oe way to solve this is by usig exactly the same techique as above but with a 9 card had istead 9 5 5 of a card had The we fid that EX 8 950 Alteratively, you ca focus o which cards ed up i the had There are 8 ways that a card had ca be draw without takig ay of the cards of rak i, so that EX 8 5 950 A card is draw from a deck of 5 Let A be the evet that the card is a heart, B be the evet that the card is red, ad C be the evet that the card is a kig Are evets A ad B idepedet? Are evets A ad C idepedet? Justify your aswer mathematically We calculate P A, P B, P C, P AB, ad P AC 5 Thus, P AB P AP B so that A ad B are ot idepedet, ad P AC P AP C, so that A ad C are idepedet 5 A ur cotais N balls, of which are blue If m balls are draw from the ur, how may of the balls draw do you expect to be blue? Let X be the umber of blue balls draw We say that X has a hypergeometric distributio with parameters, N, ad m There are two ways to do this: Way : Let X X + + X m where X i is if the ith ball draw is blue ad The EX i P X i N, so that EX m N m N Way : Label the blue balls b,, b Let X Y + + Y where Y i is if the ball b i is draw ad There are N m ways to draw m balls i additio to the ball bi, so EY i P Y i N m m m m N N, so that EX N m N 6 A red die is rolled ad the umber that comes up is observed The, a blue die is rolled that may times What is the probability that the blue die will come up 6 at least oce? P at least oe 6 o a blue die P o 6 o ay blue die 6 5 6 + 6 5 6 + + 6 5 6 6 057 7 A poit Q is chose at radom from iside the triagle with vertices at 0, 0,,, ad, 0 Let X be the x-coordiate of Q What are the expected value ad stadard deviatio of X?
Note that values of x for which more y values are possible will be more likely All x values are i the iterval [0, ] ad the value x is most likely, so X has a triagular distributio see p 97 Istead of usig those formulas, however, let s just calculate it The desity fuctio hx if 0 < x < of X is fx h x if < x < where h is chose so that the itegral of f is equal to This triagle has area, so h The EX 0 x xdx + x xdx We calculate that EX 0 x xdx + x xdx, so that σx 08 8 Suppose that the time betwee arrivals at a certai store is expoetially distributed with parameter λ 0 miutes If it s curretly :00 am, whe would the maager expect the ext customer to eter? If it turs out that o customers eter betwee :00 am ad :5 am, whe would the maager expect the ext customer to eter? We kow that the expected value of the expoetial distributio is λ, so we d expect the ext customer i 0 5 miutes that is, at :05 If it turs out that o customer has arrived by :5, the we d expect the ext customer at :0 sice the expoetial distributio is memoryless 9 I a certai regio, it rais o average oce i every te days durig the summer Rai is predicted o average for 85% of the days whe raifall actually occurs, while rai is predicted o average for 5% of the days whe it does ot rai Assume that rai is predicted for tomorrow What is the probability of raifall actually occurrig o that day? Let H be the evet that it rais ad H be the evet that it does t rai Let E be the evet that rai is predicted The, by Bayes Rule i Odds Form, P H E P H E 085 05 0 9 0 078 Covertig back to probabilities, P H E 078 +078 07 So, because rai is ucommo i this regio, it usually does t rai eve whe rai is predicted Harder Problems for those who fid the above easy 0 A fair coi is flipped +m times, ladig heads times ad tails m times After each flip, a H or a T is writte dow, so that afterward there is a record of the form HHTHTHHTTTH We call oe or more flips comig up the same i a row a ru so that, for example, the record HHTHTHHTTTH cotais 7 rus: HH, T, H, T, HH, TTT, ad H What is the expected umber of rus? Hit: Compute the umber of heads-rus ad tailsrus separately ad add these together at the ed Note that the ith flip for i begis a heads-ru if the ith flip is a heads ad the i st flip is a tails The first flip begis a heads-ru if it is a heads ad begis a tails-ru otherwise Let X be the umber of rus ad let X X H + X T where X H is the umber of headsrus ad X T are the umber of tails-rus Let X H H + + H +m where H i is if 5
the ith flip begis a heads-ru ad We compute E[H ] +m ad E[H i] P tails i positio i ad heads i positio i for i So, EX H +m + + m m A similar calculatio gives EX T +m+m m +m +m m +m + m +m, so that EX EX H+EX T + m +m You kow that bowl A has three red ad two white balls iside ad that bowl B has four red balls ad three white balls Without your beig aware of which oe it is, oe of the bowls is radomly chose ad preseted to you Blidfolded, you must pick two balls out of the bowl You may proceed accordig to oe of the followig strategies: a you will choose ad replace ie, you will replace your first ball ito the bowl before choosig your secod ball b you will choose two balls without replacig ay ie, you will ot replace the first ball before choosig a secod The blidfold is the removed ad the colors of both of the balls you chose are revealed to you Thereafter you must make a guess as to which bowl your two balls came from For each of the two possible strategies, determie how you ca make your guess depedig o the colors you have bee show Which strategy offers the higher probability for a correct guess as to which bowl the balls came from? Make a chace tree for each of the two strategies Let RR be the evet that both balls draw are red, W W be the evet that both are white, ad RW be the evet that oe ball is red ad oe is white i either order Let A be the evet that we are pickig from bowl A I the case that we replace the balls, we fid from the chace tree that P A RR 05, P A RW 095, ad P A W W 065, so that A is more likely if we draw two reds ad less likely otherwise So, the best strategy i this case is to guess that the bowl is A if we draw two red balls ad to guess that the bowl is B otherwise The the probability of wiig is P guess A correctly + P guess B corectly 9 50 + +9 98 057 I the case that we do t replace the balls, we fid from the chace tree that P A RR 05, P A RW 05, ad P A W W 0, so i this case the best strategy is to guess B if we draw two white balls ad to guess that the bowl is A otherwise I this case, the probability of wiig is ++ 0 + 05 So, the best strategy is to ot replace the balls ad to guess bowl A uless we draw two white balls Suppose that A flips a fair coi + times ad B flips a fair coi times What is the probability that A will have more heads tha B? Hit: Coditio o who has more heads after the first flips We coditio o who has more heads after the first flips Let B be the evet that A has more heads, B be the evet that A ad B are tied, ad B be the evet that B has more heads Let A be the evet that A has more heads tha B after the + st flip Iterestigly, we ca solve this problem without actually calculatig P B i for ay i Note that P B P B, so if we let P B a ad P B b, the a + b The, P A P A B P B +P A B P B +P A B P B a+ b+0 a a+b 6
If you do wat to calculate these probabilities, ote that b P B k0 P both A ad B have exactly k heads k k k k k0 k k It ca be show that this last sum simplifies to The a b 7