IE 230 Seat # Name < KEY > Please read these directions. Closed book and notes. 60 minutes.

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1 IE 230 Seat # Name < KEY > Please read these directios. Closed book ad otes. 60 miutes. Covers through the ormal distributio, Sectio 4.7 of Motgomery ad Ruger, fourth editio. Cover page ad four pages of exam. Page 8 of the Cocise Notes. No calculator. No eed to simplify beyod probability cocepts. For example, usimplified factorials, itegrals, sums, ad algebra receive full credit. Throughout, f deotes probability mass fuctio or probability desity fuctio ad F deotes cumulative distributio fuctio. A true-or-false statemet is true oly if it always true; ay couter-example makes it false. For oe poit, write your ame eatly o this cover page ad circle your family ame. Score Exam #2, March 8, 2011 Schmeiser

2 Closed book ad otes. 60 miutes. For statemets a g, choose true or false, or leave blak. (three poits if correct, oe poit if left blak, zero poits if icorrect) (a) T F The cotiuous-uiform family of distributios is a special case of the discrete-uiform family. (b) T F Scheduled arrivals, such as to physicia s office, are aturally modeled with a Poisso process. (c) T F Cosider a discrete radom variable X with probability mass fuctio f X. The f X (c ) dc = 1. (d) T F Cosider a radom variable X with Poisso distributio with mea µ=3 arrivals. Becauseσ X 2 =µ, the uits ofσ X are arrivals 1/2. (e) T F If Z is a stadard-ormal radom variable, the P(Z = 0) = 0. (f) T F If Z is a stadard-ormal radom variable, the f Z (0)=0, where f Z is the probability desity fuctio of Z. (g) T F If Z is a stadard-ormal radom variable, the F Z ( 3.2)=F Z (3.2), where F Z is the cdf of Z. 2. Suppose that the radom variable X has the discrete uiform distributio over the set {1, 2,..., 10} ad that the radom variable Y has the cotiuous uiform distributio over the set [1, 10]. For statemets a e, choose true or false. (a) (3 poits) T F E(X )=E(Y ). (b) (3 poits) T F V(X )=V(Y ). (c) (3 poits) T F F X (5.5)=F Y (5). (d) (3 poits) T F f X (5.5)= f Y (5). (e) (3 poits) T F P(X = 5)= f X (5). Exam #2, March 8, 2011 Page 1 of 4 Schmeiser

3 3. (from Motgomery ad Ruger, 4 50) For a battery i a laptop computer uder commo coditios, the time from beig fully charged util eedig to be recharged is ormally distributed with mea 260 miutes ad a stadard deviatio of 50 miutes. (a) (8 poits) Sketch (well) the correspodig ormal pdf. Label ad scale both axes. Sketch the usual bell curve, with ceter at 260 miutes ad poits of iflectio at 210 ad 310 miutes. Label the horizoal axis with a dummy variable, such as x. Label the vertical axis with f X (x ). Scale the horizoal axis with at least two umbers, such as 260 ad 310. Scale the vertical axis with at least two umbers, such as zero ad the mode height, which is 1/( 2πσ) (0.4) / (50)=0.008 (b) (6 poits) I your sketch, show the probability that a radomly selected recharge is less tha three hours. (c) (5 poits) State the umerical value of the probability i Part (b). State as much precisio as you ca determie (give that you do t have access to a ormal table). For Part (b) shade the area uder f X to the left of 180 miutes. That probability is P(X < 180)=P(Z < ( ) / 50)=P(Z 1.6) We kow that P(Z 2) 0.05 ad P(Z 1) Therefore, P(X < 180) 0.1 (d) (4 poits) Give at least oe reaso why the time util recharge caot possibly have a ormal distributio. The rage of the ormal distributio is (,). The rage of time to recharge is (0,). Exam #2, March 8, 2011 Page 2 of 4 Schmeiser

4 4. Cosider the biomial pmf f X (x )=C x p x (1 p) x for x=0, 1,..., ad zero elsewhere. (a) (5 poits) Explai, i words, the origis of p x. Iclude ay assumptios that are required for your explaatio. The probability of x idepedet trials all beig successful, whe p is the probability of success for each trial. (b) (3 poits) Evaluate C 8 3. C 8 3 = 8! = = 56 3! 5! 3 2 (c) (5 poits) Explai, i words, the phrase "ad zero elsewhere". The pmf f X is defied for every real umber x. The iterestig part of f X, those values of x that are possible, are defied explicitly. The uiterestig part of f X, those values that are impossible, are defied implicitly with the phrase "ad zero elsewhere". 5. Cosider Questio 1, which is composed of seve true-false questio. Suppose that a clueless studet is takig this exam ad aswers all true-false questios by flippig a coi, with heads yieldig "true" ad tails yieldig "false". Let X deote the umber of questios aswered correctly. (a) (6 poits) Choose a appropriate distributio for X. Iclude the family ame, parameter values, ad probability mass-or-desity fuctio f X. biomial with = 7 ad p = 0.5 The pdf is f X (x )=C x p x (1 p) x for x = 0, 1,..., ad zero elsewhere. (b) (3 poits) Cosider aother clueless studet who leaves all seve questios blak. Determie this studet s expected umber of poits? The mea is 7 poits, sice the studet always gets 7 poits. (c) (3 poits) Agai cosider the studet who leaves all seve questios blak. Determie the stadard deviatio of this studet s umber of poits. The std is zero poits, sice always the studet always receives seve poits. Exam #2, March 8, 2011 Page 3 of 4 Schmeiser

5 6. (from Motgomery ad Ruger, 4 11) Suppose that the cumulative distributio fuctio of the radom variable X is F X (x )=0.2x for 0 x c. Assume that values outside the iterval [0, c ] are ot possible. (a) (5 poits) Sketch F X over the etire real-umber lie. Label ad scale both axes. Sketch two perpedicular axes. Label the horizotal axis with a dummy variable, such as x. Label the vertical axis with F X (x ). Scale the horizotal axis with at least two umbers, probably zero ad c. Scale the vertical axis with at least two umbers, probably zero ad oe. Plot F X, showig the values for all real umbers x. (b) (5 poits) Determie the value of c. F X (c )=F X (c )=1 at the upper boud, so c = 5 (c) (5 poits) Write f X. Be complete. The desity fuctio is the first derivative of the cdf, so f X (x )=1 / 5 for 0 x 5 ad is zero elsewhere. Exam #2, March 8, 2011 Page 4 of 4 Schmeiser

6 Discrete Distributios: Summary Table radom distributio rage probability expected variace variable ame mass fuctio value X geeral x 1, x 2,..., x P(X = x ) x i f (x i ) (x i µ) 2 f (x i ) = f (x ) =µ=µ X =σ 2 =σ X 2 = f X (x ) = E(X ) = V(X ) = E(X 2 ) µ 2 X discrete x 1, x 2,..., x 1 / x i / [ x i 2 / ] µ 2 uiform X equal-space x = a,a+c,...,b 1 / a+b c 2 ( 2 1) 2 12 uiform where = (b a+c ) / c "# successes i idicator x = 0, 1 p x (1 p ) 1 x p p (1 p ) 1 Beroulli variable trial" where p = P("success") "# successes i biomial x = 0, 1,..., C x p x (1 p ) x p p (1 p ) Beroulli trials" where p = P("success") "# successes i hyper- x = C K x C 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 replacemet) K successes" "# Beroulli geometric x = 1, 2,... p (1 p ) x 1 1 / p (1 p ) / p 2 trials util 1st success" where p = P("success") "# Beroulli egative x = r, r+1,... C r 1 p r (1 p ) x r r / p r (1 p ) / p 2 trials util biomial r th success" where p = P("success") "# of couts i Poisso x = 0, 1,... e µ µ x / x! µ µ time t from a Poisso process where µ = λt with rateλ" Result. For x = 1, 2,..., the geometric cdf is F X (x )=1 (1 p ) x. Result. The geometric distributio is the oly discrete memoryless distributio. That is, P(X > x + c X > x )=P(X > c ). Result. The biomial distributio with p = K / N is a good approximatio to the hypergeometric distributio whe is small compared to N. Purdue Uiversity 5 of 22 B.W. Schmeiser

7 IE230 CONCISE NOTES Revised August 25, 2008 Cotiuous Distributios: Summary Table radom distributio rage cumulative probability expected variace variable ame distrib. 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 ] x a 1 a + b (b a ) 2 b a b a 2 12 uiform (x a )f (x ) / 2 X triagular [a, b ] 2(x d ) a+m+b (b a ) 2 (m a )(b m ) if x m, else (b a )(m d ) (b x )f (x )/2 (d = a if x m, else d = b ) 1 x µ 2 e 2 σ sum of ormal (,) Table III µ σ 2 2πσ radom (or variables Gaussia) time to expoetial [0,) 1 e λx λ e λx 1 /λ 1 /λ 2 Poisso cout 1 time to Erlag [0, ) e λx (λx ) k λ r x r 1 e λx r /λ r /λ 2 k! (r 1)! k=r Poisso cout r lifetime gamma [0, ) umerical λ r x r 1 e λx r /λ r /λ 2 Γ(r ) lifetime Weibull [0,) 1 e (x βx β 1 (x /δ)β e /δ)β δγ(1+ 1 ) δ 2 Γ( ) µ δ β β β Defiitio. For ay r > 0, the gamma fuctio isγ(r )= x r 1 e x dx. 0 Result. Γ(r ) = (r 1)Γ(r 1). I particular, if r is a positive iteger, the Γ(r ) = (r 1)!. Result. The expoetial distributio is the oly cotiuous memoryless distributio. That is, P(X > x + c X > x )=P(X > c ). Defiitio. A lifetime distributio is cotiuous with rage [0,). Modelig lifetimes. Some useful lifetime distributios are the expoetial, Erlag, gamma, ad Weibull. Purdue Uiversity 12 of 22 B.W. Schmeiser

Closed book and notes. No calculators. 60 minutes, but essentially unlimited time.

Closed book and notes. No calculators. 60 minutes, but essentially unlimited time. IE 230 Seat # Closed book ad otes. No calculators. 60 miutes, but essetially ulimited time. Cover page, four pages of exam, ad Pages 8 ad 12 of the Cocise Notes. This test covers through Sectio 4.7 of