Closed book ad otes. No calculators. 120 miutes. Cover page, five pages of exam, ad tables for discrete ad cotiuous distributios. Score X i =1 X i / S X 2 i =1 (X i X ) 2 / ( 1) = [i =1 X i 2 X 2 ] / ( 1) E(W ) ceter of gravity of f W COV(X, Y ) E[ (X E(X )) (Y E(Y )) ] = E( XY ) E(X )E(Y ) V(X ) COV(X, X ) CORR(X, Y ) COV(X, Y ) / V(X )V(Y ) E(aX + by ) = a E(X ) + b E(Y ) V(aX + by ) = a 2 V(X ) + b 2 V(Y ) + 2ab COV(X, Y ) Fial Exam (a), Sprig 2001 Schmeiser
1. True or false. (for each, 2 poits if correct, 1 poit if left blak.) (a) T F The formula for sample variace, s 2 i =1 (x i x )2 / ( 1), applies oly whe the observatios x i are from a cotiuous distributio. (b) T F "Maximum-likelihood estimatio" determies the sample of observatios that is most likely for a give assumed distributio. (c) T F I iferetial statistics, coclusios about a populatio arise from observig a radom sample. (d) T F If X ad Y are idepedet, the COV(X, Y ) = 0, regardless of whether the radom variables X ad Y are cotiuous. (e) T F If X ad Y are cotiuous, the COV(X, Y ) = 0, regardless of whether the radom variables X ad Y are idepedet. (f) T F The liear combiatio ax+ by is a discrete radom variable regardless of whether X ad Y are cotiuous or discrete. (g) T F If X is a idicator radom variable, the E(X ) is a probability. (h) T F All ormal distributios differ oly i locatio ad scale; that is, their desity fuctios all have the same shape. (i) T F Although ot useful i practice, a 100% cofidece iterval for ay distributio parameter θ is the real-umber lie, (, ). 2. Result: E(X + Y ) = E(X ) + E(Y ). Assume that X ad Y are cotiuous. Prove the result, providig a reaso for each step. Fial Exam (a), Sprig 2001 Page 1 of 5 Schmeiser
3. Let T i, i = 1, 2, 3, deote the score o test i of a radomly selected studet i IE230 this semester. Assume that T i is ormally distributed with mea µ i ad variace σ i 2. (a) Sketch the pdf of T 2. Label ad scale the horizotal axis. Label the vertical axis. (b) Sketch the cdf of T 2. Label ad scale the horizotal axis. Label ad scale the vertical axis. (c) Assume that the three test scores are idepedet. Fid the probability that all of three test scores are more tha oe stadard deviatio above the mea. (d) The idepedece assumptio i Part (c) is ot good (because someoe who scores well o oe test is likely to score well o other tests). Therefore, the aswer to Part (c) will be (circle oe) (i) too high (ii) too low (iii) either Fial Exam (a), Sprig 2001 Page 2 of 5 Schmeiser
4. Suppose that 50% of a compay s employees are uder 30 years old ad that 40% are wome. Also suppose that 10% of the employees are wome uder 30. If a employee is chose at radom, what is the probability that the chose employee is a male 30 years or older? 5. Assume that the time that a studet speds gettig food before arrivig at the cashier at Purdue s Uio Market is expoetially distributed with mea 3 miutes. (a) What fractio of the studets sped more tha five miutes? (b) You ad a fried eter at the same time. Three miutes later you arrive at the cashier, where you otice that your fried has ot yet arrived. You decide to wait for your fried to arrive. What is the pdf of the time that you will have to wait? Fial Exam (a), Sprig 2001 Page 3 of 5 Schmeiser
6. (Motgomery ad Ruger, 3 82) Customers are used to evaluate prelimiary product desigs. I the past, 90% of highly successful products received good reviews, 50% of moderately successful products received good reviews, ad 10% of poor products received good reviews. I additio, 45% of products have bee highly successful, 30% have bee moderately successful, ad 25% have bee poor products. (a) What is the probability that a product receives a good review? (b) If a ew desig receives a good review, what is the probability that it will be a highly successful product? (c) If a product does ot receive a good review, what is the probability that it will be a highly successful product? Fial Exam (a), Sprig 2001 Page 4 of 5 Schmeiser
7. A multiple-choice exam has 100 questios, each with five possible aswers. Each questio is worth oe poit. (a) Suppose that a studet guesses radomly. What is the distributio of the studet s score? (b) Suppose that the studet ca elimiate oe choice for each questio. The studet the guesses for each of the remaiig four choices. I terms of expected value, how may poits better off is the studet for havig elimiated the oe choice? (c) Suppose that you aswer 85 questios correctly. Also suppose that, to save time, oly 20 radomly chose questios are graded. What is the probability mass fuctio of the umber of graded questios that you have correct? Sprig 2001 Page 5 of 5 Schmeiser
Discrete-Distributios: Summary Table (from the Cocise Notes) radom distributio rage probability expected variace variable ame mass fuctio value X geeral x 1, x 2,...,x P(X = x ) i =1 x i f (x i ) i =1 (x i µ) 2 f (x i ) = f (x ) =µ=µ X =σ 2 2 =σ X = f X (x ) = E(X ) = V(X ) = E(X 2 ) µ 2 X discrete x 1, x 2,...,x 1 / uiform X equal-space x = a,a +c,...,b 1 / uiform where = (b a +c ) /c i =1 a +b 2 x i / [ i =1 x i 2 /] µ 2 c 2 ( 2 1) 12 "# successes i biomial x = 0, 1,..., C x p x (1 p ) x p p (1 p ) Beroulli trials" "# Beroulli geometric x = 1, 2,... p (1 p ) x 1 1 /p (1 p ) /p 2 trials util 1st success" "# Beroulli egative x = r, r +1,... x C r 1 p r (1 p ) x r r/p r(1 p ) /p 2 trials util biomial r th success" "# successes i hyper- x = C K N x C K N x /C p p (1 p ) (N ) (N 1) a sample of geometric ( (N K )) +, size from..., mi{k, } where p = K/N a populatio (Samplig ad of size N without iteger cotaiig k successes" replacemet) "# of couts i Poisso x = 0, 1,... e µ µ x /x! µ µ a Poissoprocess iterval" Sprig 2001 Page 1 of 2 Schmeiser
Cotiuous-Distributios: Summary Table radom distributio rage cumul. probability expected variace variable ame dist. fuc. desity fuc. value X geeral (, ) P(X x ) df (y ) dy y =x xf (x )dx (x µ) 2 f (x )dx = F (x ) = f (x ) =µ=µ X =σ 2 2 =σ X = F X (x ) = f X (x ) = E(X ) = V(X ) = E(X 2 ) µ 2 X cotiuous [a, b ] uiform x a b a sum of ormal (, ) Table II radom variables (or Gaussia) 1 b a 1 x µ 2 e 2 σ a + b 2 µ σ 2 2π σ (b a ) 2 time to expoetial [0, ) 1 e λx λ e λx 1 / λ 1 / λ 2 Poisso cout 12 time to r th Erlag [0, ) Poisso cout k =r e λx (λx ) k k! λ r x r 1 e λx r/λ r/λ 2 (r 1)! lifetime gamma [0, ) umerical λ r x r 1 e λx r/λ r/λ 2 Γ(r ) βx β 1 e (x/δ)β lifetime Weibull [0, ) 1 e (x/δ)β δγ(1+ 1 ) δ 2 Γ(1+ 2 ) µ 2 δ β β β Sprig 2001 Page 2 of 2 Schmeiser