Sample questions. 8. Let X denote a continuous random variable with probability density function f(x) = 4x 3 /15 for

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1 Sample questios Suppose that humas ca have oe of three bloodtypes: A, B, O Assume that 40% of the populatio has Type A, 50% has type B, ad 0% has Type O If a perso has type A, the probability that they have hemophelia is 005 People with Type B have a 003 probability of havig hemophelia, ad people with Type O have a 00 chace of carryig hemophilia If a perso is selected at radom from this populatio: (a) What is the probability that the perso has hemophilia? (b) If the perso has hemophilia, what is the probability that they have Type A blood? A bi cotais 8 light bulbs, 6 of which work properly, but the other are defective You select for purchase 3 bulbs at radom ad without replacemet from the bi (a) What is the probability that you get two defective bulbs? (b) What is the probability that you get at least oe defective bulb? 3 The umber of pulses a commuicatios chael receives per secod ca be modeled with a Poisso distributio with a rate parameter of 0 (a) What is the probability the chael receives at least pulses i a oe secod iterval? (b) What is the probability that o pulses are received i a 5 secod iterval? 4 Let A ad B deote two evets i a sample space You are give that P(A)=03, PA ( B) 08 ad that A ad B are idepedet evets Determie P(B) 5 There are 5 people i a hospital sufferig from a particular disease The drug they take has a 0% success rate; that is, each patiet has a 00 probability of recovery Assumig that the patiets' fates are idepedet of oe aother: (a) What is the probability that 4 or more patiets recover? (b) What is the probability that at least oe patiet recovers? 6 A stadard 5 card deck cotais 3 cards (ace,,3,4,5,6,7,8,9,0, jack, quee ad kig) from each of 4 suits (spades, clubs, diamods, hearts) (a) If you are dealt 4 cards without replacemet, what is the probability of tettig spades, a heart, ad a club? (b) If you repeat the experimet 00 times, each time returig the 4 cards to the deck, what is the probability of gettig 4 kigs at least oe time i the 00 hads? 7 I a previous versio of the state lottery, the perso playig the lottery selected without replacemet 6 umbers from a set of 45 umbers {0,,,,44} At some time later a computer radomly selected without replacemet 6 umbers from this same set ad you wo the big prize if you matched all 6 umbers the computer selected If your strategy was to buy a ticket for each possible outcome, how may tickets would you have to purchase? 8 Let X deote a cotiuous radom variable with probability desity fuctio f(x) = 4x 3 /5 for x (a) Determie the probability that X > 5

2 (b) Determie the cumulative distributio fuctio F(x) ad state the values of F(x) at x = 05, 5, ad 5 (c) Fid the desity fuctio of Y = X Be sure to state the domai of f(y) 9 The amout of cereal i a box is ot costat, but the distributio ca be modeled with a Gaussia distributio with a mea of 65 ouces If the maufacturer is required to fill 90% of the cotaiers with 6 (or more) ouces of cereal, what is the largest allowable value of the stadard deviatio, s? 0 The umber of pizza orders received at a pizza place follows a Poisso model with a mea rate of 7 per hour (a) What is the probability that the pizza shop goes more tha /hour betwee orders? (b) If it has bee hour sice the last order, what is the probability that a order arrives i less tha 5 miutes? A pizza shop makes deliveries, ad the time to make the delivery follows a uiform distributio betwee 0 ad 35 (miutes): f(x) = /5 for 0 < x < 35 (a) Fid the average delivery time ad the stadard deviatio of the delivery times (b) Accordig to Chebyshev's theorem, at least 75% of the delivery times must be betwee what two values? (c) O each trip, the supervisor of the drivers gives a bous of $00 for each miute below 35 For example, if a driver takes 8 miutes, that is a $070 bous What is the average bous per trip? The lifetime of a lightbulb follows a Gaussia model with a mea of 000 hours ad a stadard deviatio of 00 hours (a) If I purchase of these bulbs, what is the probability the bulb lasts at least 00 hours? (b) If I purchase 4 of these bulbs, what is the probability at all four of them lasts more tha 00 hours? (c) I purchased a bulb that has bee operatig for 900 hours What is the probability it lasts aother 300 hours? 3 The cotiuous radom variable X has probability desity fuctio f(x) = for 0 < x < 0 (a) Determie the probability desity fuctio g(y) of Y l( X) Be sure to state the domai of g(y) (b) Determie the expected value of the fuctio e X (This is e, the base of the atural logarithms, raised to the X power) 4 Suppose X ad Y are idepedet radom variables, each followig a Gaussia distributio The parameters are: 0, 5, 9, 6 (a) Compute P(X < 3, Y > ) (b) Compute P( X + Y > 48 ) x y x y (c) Give that X ad Y are idepedet, what is the expected value of the quatity (XY), E(XY)?

3 5 The joit desity fuctio of (X,Y) is f(x,y) = x/for 0 < y < x < (a) Compute P(X < ) (b) Compute P( Y < X = 5 ) (c) Compute P(X<0, Y<05) 6 The joit desity fuctio of (X,Y) is f(x,y) = x/ for 0 < y < x < You are give that E(Y) = /, Var(Y) = 7/36, ad that E(X) = 0 Fid the correlatio betwee X ad Y 7 You geerate 00 idepedet observatios form the pdf f(x) = /4for 0 < X < 4 What is the (approximate) probability that the sum of these 00 observatios is less tha 90? 8 The time to failure of a device follows the probability model f(x) = 0/x for x > 0 (X is measured i moths) 9 You have 3 such devices operatig idepedetly What is the probability that all 3 survive at least 0 moths? 0 You have two such devices operatig idepedetly Let X ad Y deote the times to failure of the two devices What is P(X > 30 Y > 0)? The fracture stregth of tempered glass average 4 (measured i thousads of pouds per square ich) ad has stadard deviatio (a) What is the probability that the average fracture stregth of 00 radomly selected pieces of this glass exceeds 45? (b) Fid a iterval that icludes, with probability 095, the average fracture stregth of 00 radomly selected pieces of this glass Let YY,,, Y be idepedet radom variables, each with a beta distributio, with Fid (a) the probability distributio fuctio of Y( ) max( Y, Y,, Y) (b) the desity fuctio of Y ( ) (c) EY ( ( ) ) for = 3 Let Y ad Y be idepedet ad uiformly distributed over the iterval (0, ) Fid the probability desity fuctio of each of the followig: U Y / Y (a) (b) U Y 3 4 Let YY,,, Y 6 be idepedet radom variables from a stadard ormal populatio ad let Y (/ 5) Yi What is the distributio of 5 i

4 5 Y i i 5 ( ) i i (a) W? Why? (b) U Y Y? Why? (c) 5 Y6 / (d) W? Why? (5 Y Y ) / U? Why? 6 5 Let YY,,, Y be idepedet Poisso radom variables with meas,,,, respectively Fid the (a) probability fuctio of Y i i (b) coditioal probability fuctio of Y, give that Y i i m 6 Resistors to be used i a circuit have average resistace 00 ohms ad stadard deviatio 0 ohms Suppose 5 of these resistors are radomly selected to be used i a circuit (a) What is the approximate probability that the average resistace for the 5 resistors is betwee 99 ad 0 ohms? (b) Fid the approximate probability that the total resistace does ot exceed 500 ohms 7 To estimate the proportio of uemployed workers i Paama, a ecoomist selected at radom 400 persos from the workig class Of these, 5 were uemployed (a) Estimate the true proportio of uemployed workers (b) Fid a 95% cofidece iterval for the proportio of uemployed workers i Paama (c) How may persos must be sampled to reduce the boud o the error of estimatio to 00 with 95% cofidece? 8 Two methods for teachig readig were applied to two radomly selected groups of elemetary schoolchildre ad were compared o the basis of a readig comprehesio test give at the ed of the learig period The sample meas ad variaces computed from the test scores are show i the accompayig table Fid a 95% cofidece iterval for ( ) What assumptios are ecessary? Statistic Method Method Number of childre i group 4 y s 5 7

5 9 If X, X,, X costitute a radom sample from the populatio give by ( x ) e for x f( x) 0 elsewhere (a) Show that ˆ X is a biased estimator of (b) What is the bias B( ˆ )? (c) Fid the MSE( ˆ ) (d) Base o ˆ i a), fid a ubiased estimator ˆ of (e) What is the MSE( ˆ )? 30 If YY,,, Y costitute a radom sample of size from a expoetial populatio with parameter (a) Fid the miimal sufficiet statistic of the parameter (b) Fid the MVUE of the parameter (c) Show that the MVUE obtaied i b) is a cosistet estimator of the parameter (d) Fid the maximum likelihood estimator of the parameter 3 Suppose that YY,,, Y costitute a radom sample from the desity fuctio ( y e ), y, f( y ) 0, elsewhere where is a ukow, positive costat (a) Fid a estimator for by the method of momets (b) Fid a estimator for by the method of maximum likelihood (c) Adjust ad so that they are ubiased Fid the efficiecy of the adjusted relative to the adjusted 3 Do you believe that a exceptioally high percetage of the executives of large corporatios are right-haded? Although 85% of the geeral public is right-haded, a survey of 300 chief executive officers of large corporatios foud that 96% were right-haded Is this differece i percetages statistically sigificat? (a) State the ull ad alterative hypotheses (b) Calculate the observed value of the appreciate test statistic (c) What is the appropriate rejectio regio for a 00 level test? (d) What is your coclusio?

6 34 Two sets of elemetary schoolchildre were taught to read by usig differet methods, 50 by each method At the coclusio of the istructioal period, a readig test yielded the results y 74, y 7, s 9, s 0 We wat to see weather evidece idicates a differece betwee the two populatio meas (a) State the ull ad alterative hypotheses (b) Calculate the observed value of the appreciate test statistic (c) What is the appropriate rejectio regio for a 00 level test? (d) What is your coclusio?