First Year Examination Department of Statistics, University of Florida


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1 Frst Year Examnaton Department of Statstcs, Unversty of Florda May 7, 010, 8:00 am  1:00 noon Instructons: 1. You have four hours to answer questons n ths examnaton.. You must show your work to receve credt. 3. Questons 1 through 5 are the theory questons and questons 6 through 10 are the appled questons. You must do exactly four of the theory questons and exactly four of the appled questons 4. Wrte your answers to the theory questons on the blank paper provded. Wrte only on one sde of the paper, and start each queston on a new page. 5. Wrte your answers to the appled questons on the exam tself. 6. Whle the 10 questons are equally weghted, some questons are more dffcult than others. 7. The parts wthn a gven queston are not necessarly equally weghted. 8. You are allowed to use a calculator. 1
2 The followng abbrevatons and termnology are used throughout: ANOVA = analyss of varance CRD = completely randomzed desgn d = ndependent and dentcally dstrbuted mgf = moment generatng functon ML = maxmum lkelhood MOM = method of moments pdf = probablty densty functon pmf = probablty mass functon UMP = unformly most powerful Z + = {0, 1,, 3,... } N = {1,, 3,... } R + = (0, ) You may use the followng facts/formulas wthout proof: Beta densty: X Beta(α, β) means X has pdf where α > 0 and β > 0. f(x; α, β) = Gamma densty: X Gamma(α, β) means X has pdf where α > 0 and β > 0. f(x; α, β) = Γ(α + β) Γ(α)Γ(β) xα 1 (1 x) β 1 I (0,1) (x) 1 Γ(α) β α xα 1 e x/β I (0, ) (x) Negatve Bnomal mass functon: X NB(r, p) means X has pmf ( ) r + x 1 P (X = x) = p r (1 p) x I x Z +(x), where p (0, 1) and r N. Also, E(X) = r(1 p)/p and Var(X) = r(1 p)/p. Iterated Expectaton Formula: E(X) = E [E(X Y )]. Iterated Varance Formula: Var(X) = E [Var(X Y )] + Var [E(X Y )].
3 1. Suppose we have two urns. Urn I contans three yellow balls, four red balls, and two green balls. Urn II contans one yellow ball, two green balls, and four purple balls. Consder a twostage experment n whch we randomly draw three balls from Urn I and move them to Urn II, and then we randomly draw one ball from (the new) Urn II. (a) Defne two events as follows: and A = { Two yellow balls and one green ball are moved from Urn I to Urn II } B = { A green ball s drawn from Urn II }. Fnd the probabltes of these two events. (Express all numercal answers n ths problem as ratos of ntegers or sums of ratos of ntegers  do not use any decmals.) (b) Are A and B ndependent? (c) Fnd the probablty that at least two of the balls moved from Urn I to Urn II were yellow, gven that the ball drawn from Urn II was yellow.. Let (X, Y ) be a bvarate random vector wth jont pdf [ ] f X,Y (x, y) = c I (0,1) (x)i ( 1,0) (y) xi ( 1,0) (x)i (0,1) (y), where c s an unknown normalzng constant. Let S R denote the support of the densty f X,Y (x, y). (a) Fnd the value of c. (b) Let A R denote the trangle whose vertces are at the ponts ( 1/, 0), (1/, 1/3) and (1/, ). Fnd P ( (X, Y ) A ). (c) Calculate the margnal pdf of X. (d) Calculate the condtonal pdf of Y gven X = x. (e) Fnd the condtonal expectaton of Y gven that X = 1/4. (f) Fnd a second jont pdf, call t f X,Y (x, y), that has exactly the same margnals as f X,Y (x, y), but whose support, call t S, has twce as large an area as S. (g) Fnd a thrd jont pdf, call t f X,Y (x, y), that has exactly the same xmargnal as f X,Y (x, y), but whose support, call t S, satsfes S S =. 3
4 3. Suppose that the condtonal pmf of Z gven θ s and that, margnally, θ Beta(α, β). (a) Fnd the margnal pmf of Z. P (Z = z θ) = θ (1 θ) z I Z +(z), (b) Derve a formula for E [ θ a (1 θ) b] that nvolves only α, β, a, b and the gamma functon. Does ths formula hold for all (a, b) R? (c) Calculate the margnal mean and varance of Z, assumng that α >. (Hnt: Use terated expectaton and varance.) For the remander of ths problem, assume that Z 1,..., Z n are d wth common pmf gven by where α >. P (Z = z; α) = α (α + 1) Γ(α) (z + 1)! Γ(α + z + 3) (d) Fnd the MOM estmator of α and call t α n = α n (z). (e) Show that and dentfy the functon v( ). n ( αn α ) d N ( 0, v(α) ), I Z +(z), 4. Let X 1,..., X n be d random varables from the pdf where θ > 0. f(x; θ) = x θ e x /θ I R +(x) (a) What knd of parameter s θ: locaton, scale or nether? (Explan your answer.) (b) Derve a formula for EX p that holds for any p >. (c) Fnd a functon of n, call t g(n), such that g(n) n =1 X4 s unbased for θ. (d) Fnd the ML estmator of θ. (e) Fnd the CramérRao Lower Bound for the varance of an unbased estmator of θ. (f) Fnd the best unbased estmator of θ. 4
5 5. Suppose that Y 1,..., Y n are d NB(r, q). (a) Derve the mgf of Y 1. (b) Derve the dstrbuton of n =1 Y. (c) Suppose we have four dentcal quarters and we would lke to test H 0 : p 1/ vs. H 1 : p > 1/, where p s the unknown probablty that any one of the quarters comes up heads when t s flpped. We decde to perform an experment. Each quarter wll be pcked up and flpped repeatedly untl the frst head occurs. Let X denote the number of tals that occur before the frst head occurs when the th quarter s flpped, = 1,, 3, 4. Use the X s to construct a UMP sze test of H 0 vs. H 1. 5
6 Q.6. Consder models: Model 1: y u u ~ NID0, 1,...,4 j 1,..., 3 Model: y v NID0 p.6.a. In matrx form, we can wrte the models as follows: u 0 1 ~, v Model 1: 1 0 Y Wα U α Model : Y Xβ V β Fll n all empty values below: y y y y y y Y y y y y y y W X W' W _ X'X _ W' W 1 1 X'X X'Y β W' Y α p.6.b. For ths data, compute: Y Wα T Y Wα _ Y Xβ T Y Xβ _ Gve the test statstc and rejecton regon for testng: H 0 : = H A : Test Statstc: Rejecton Regon (based on 0.05 sgnfcance level): Do you reject the null hypothess that the mean s lnearly related to treatment level ()? Yes / No
7 Q.7. Derve the expected mean squares for treatments and error for the balanced 1Way ANOVA wth g g treatments and n replcates per treatment: y 0 ~ NID0, 1
8 Q.8. A multple regresson model s ft based on n=30 ndvduals, relatng Y to X 1 and X (the model also contans an ntercept). The coeffcent of determnaton s R = Note: Y Y. p.8.a. Complete the Analyss of Varance and test H 0 : at the 0.05 sgnfcance level Source df SS MS F F(.05) Regresson Resdual Total p.8.b. Based on the least squares estmate of the parameter vector, and (X X) 1 gven below, test H 0 : at the 0.05 sgnfcance level (X'X)(1) Betahat
9 Q.9. An experment s conducted as a Randomzed Complete Block Desgn wth 3 treatments appled to 8 blocks. The model ft s gven below, as well as the data: y j 4 0, ~ NID0, COV, 0, j, j' 1,...,3 j 1,...,8 0 j ~ NID j' 1 Trt1 Trt Trt3 Mean Block Block Block Block4 0 4 Block Block Block Block Mean y y 688 p.9.a. Complete the followng ANOVA table, and test whether treatment effects exst at the 0.05 sgnfcance level. H 0 : Source df SS MS F F(.05) Trts Blocks Error Total p.9.b. Use Bonferron s method to obtan smultaneous 95% confdence ntervals for all pars of dfferences among treatment means. 1 j1
10 Ths queston s based on the followng regresson model, and X s of full column rank (no lnear dependences among predctor varables). Y Xβ ε X n p' β p' 1 ε ~ N0, I Gven: a' x dx'ax d dx a dx. For a symmetrc matrx A: Ax E Y' AY travy μ Y ' Aμ Y Cochran s Theorem: Suppose Y s dstrbuted as follows wth nonsngular matrx V: Y ~ N μ, V rv n then f AV s dempotent : 1 1 Y' A Y sdstrbuted non  central wth : (a) df ( A) and (b) Noncentralty parameter : r Q.10.a. Derve the least squares estmator for μ' Aμ n n SS(Model) 1 SS(Resdua l) 1 Q.10.b. Obtan Y and e n matrx form (quadratc forms n Y). 1 1 Obtan the dstrbutons of the two sum of squares (be specfc wth regard to ther famly of dstrbutons, degrees of freedom, and noncentralty parameters).
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