2. The volume of the solid of revolution generated by revolving the area bounded by the
|
|
- Nigel Webb
- 5 years ago
- Views:
Transcription
1 IIT JAM Mathematical Statistics (MS) Solved Paper. A eigevector of the matrix M= ( ) is (a) ( ) (b) ( ) (c) ( ) (d) ( ) Solutio: (a) Eigevalue of M = ( ) is. x So, let x = ( y) be the eigevector. z (M I)X = x ( ) ( y) = ( ) z y =, z = Hece ( ) is a eigevector.. The volume of the solid of revolutio geerated by revolvig the area bouded by the curve y = x ad the straight lies x = 4 ad y = about the x axis, is (a) π (b) 4π (c) 8π (d) π Solutio: (c) Volume of solid revolutio about x axis 4 v = π y dx = πxdx = πx 4 = 8π. 4
2 x x y y y y 3. Let I = xydxdy. The chage of order of itegratio i the itegral gives I as y y y y (a) I = xydxdy + xydxdy (b) I = xydxdy + xydxdy (c) I = xydxdy + xydxdy (d) I = xydxdy + xydxdy Solutio: (a) I the give itegratio, regio bouded is from y = x to y = x ad x = to x =. By chagig the order of itegratio, we get y y xydxdy + xydxdy y= x= y= x= 4. Let L = lim [f ( ) + f ( ) + + f (k ) kf()] where k is a positive iteger, if f(x)= si x, the L is equal to (a) (k+)(k+) 6 (b) (k+)(k+) (c) k(k+) (d) k(k + ) Solutio: (c) L = lim [f ( h ) + f ( h ) + + f (k ) kf()] = lim [si h + si h + + si k k si ]
3 L = lim si h h + si h L = k = + + si k k k(k + ) k 5. Let f(x, y) = { x + y si ( x + y ) if (x, y) (, ) if (x, y) = (, ) The at the poit (, ) (a) f is cotiuous ad f f ad exist. x y (b) f is cotiuous ad f f ad do ot exist. x y (c) f is ot cotiuous ad f f ad exist. x y (d) f is ot cotiuous ad f f ad do t exist. x y Solutio : (b) So, f(x,y) is cotiuous at (,) Which does ot exist. Lt f(x, y) = Lt f(r cos θ, r si θ) (x, y) (,) r = Lt r r si r = = f(,) t at (,) = Lt f( + h, ) f(,) x h h t x = Lt h h si h h f(,) + k f(,) at (,) = lim k k k si k = lim does ot exist. k k
4 6. Let {a } be a real sequece covergig to a, where a >. The (a) a coverges but a diverges (b) a diverges but a coverges (c) Both a ad a coverge (d) Both a ad a diverge Solutio: (d) As, lim a = a > So, = a diverges as lim a = is ecessary coditio for covergece. a Also = diverges as lim ( a ) = a > ad by Śleszyński Prigsheim theorem lim u = is ecessary for covergece of positive term series. 7. A four digit umber is chose at radom. The probability that there are exactly two zeroes i that umber is (a).73 (b).973 (c).7 (d).7 Solutio: (c) Number of 4 digit umbers = 9 3 = 9 Number of 4 digit umbers with exactly zeroes = 9C 3C 9 = 43 So, required probability = 43 9 =.7 8. A perso makes repeated attempts to destroy a target. Attempts are made idepedet of each other. The probability of destroyig the target i ay attempt is.8. Give that he fails to destroy the target i the first five attempts, the probability that the target is destroyed i the 8 th attempt is (a).8 (b).3 (c).6 (d).64 Solutio: (b) Probability that target will be hit i 8 th attempt is equal to product of probability of failure i the 6 th ad 7 th attempt ad success i the 8 th attempt. P = (.)(.)(.8) =.3
5 9. Let the radom variable X~B(5, p) such that P(X = ) = P(X = 3). The the variace of X is (a) /3 (b) /9 (c) 5/3 (d) 5/9 Solutio: (b) P(x = ) = P(x = 3) ( 5 ) p ( p) 3 = 5 c3 p 3 ( p) Variace of x = p( p) = = 3. p = p 3p = p = 3. Let X,., X 8 be i.i.d. M(, σ ) radom variables. Further, let U = X + X ad V = (a) /8 (b) ¼ (c) ¾ (d) ½ 8 i= X i. The correlatio coefficiet betwee U ad V is Solutio: (d) E(X i ) = ; V(X i ) = σ ; E(X i ) = σ ; E(UV) = E(X + X ) = σ E(U)E(V) = ; Var (U) = σ ; Var (V) = 8σ ; Cov (U, V) = E(UV) E(U)E(V) = σ Correlatio coefficiet U ad V is = COV (U, V) var (U), Var (V) = σ σ, 8σ =. Let X~F 8.5 ad Y~F 5.8. If P(X>4) =.ad P(Y k) =., the the value of k is (a).5 (b).5 (c) (d) 4 Solutio: (b) X~F 8, 5 ad Y~F 5, 8 Y = X ; P(X > 4) = P ( ) = P (Y )= P(Y.5). X 4 4
6 . Let X,, X be i.i.d. Exp () radom variables ad S = i= X i. Usig the cetral limit theorem, the value of lim P(S > ) is (a) (b) /3 (c) ½ (d) Solutio: (c) X i ~ exp() E(X i ) = ; V(X i ) = Where σ is v(x i ) S μ ξ, where ξ follows N(, σ ) lim P(S > ) = lim P ( S. > ) l N(,)P(x > ) = / 3. Let the radom variable X~U (5, 5 + θ). Based o a radom sample of size, say X, the ubiased estimator of θ is (a) 3(X 5) (b) X 5 (c) 3(X + 5) (d) X +5 Solutio: (a) 5+θ E(X ) = x. θ dx = (5 + θ) θ So, 3(X 5) is ubiased estimator of θ. 3θ = θ 3 + 5θ + 5 E(X) = x. θ dx = (5 + θ) 5 = θ θ E(X 5) = E(X ) E(X) + 5 = θ 3 E[3(X 5) ] = θ
7 4. Let X,, X be a radom sample of size from N(μ, 6) populatio. If a 95% cofidece iterval for μ is [X. 98, X +. 98], the the value of is (a) 4 (b) 6 (c) 3 (d) 64 Solutio: (d) 95% cofidece iterval for μ is [X.96σ, X +.96σ ].96σ =.98 = σ = (4) = 8 = A coi is tossed 4 times ad p is the probability of gettig head i a sigle trial. Let S be the umber of head(s) obtaied. It is decided to test H : p = agaist H : p Usig the decisio rule: Reject H if S is to 4. The probabilities of Type I error (α), ad Type II error (β) whe p = 3 4, are (a) α = 87, β = 4 8 (b) α = 87, β = 8 8 (c) α = 4, β = 8 56 (d) α = 4, β = 4 56 Solutio: (b) α = p (reject H /H is true) i.e., p = / & Rejectio H meas S =,4 α = P(S = or 4 P = /) = 4C ( 4 ) + 4C 4 ( 4 ) = 8 β = P(accept H H is true) = P(S =,,3 P = 3/4) = 4C ( 3 4 ) ( 4 ) 3 = + 4C ( 3 4 ) ( 4 ) + 4C 3 ( 3 4 ) 3 ( 4 ) = = 87 8.
8 6. (a) Fid the value(s) of λ for which the followig system of liear equatios λ x ( λ ) ( y) = ( ) λ z (i) (ii) (iii) Has a uique solutio Has ifiitely may solutios Has o solutio (b) Let a =, b = ad for, a + = a + b, b + = a b a + b Show that (i) b a, for all (ii) b + b for all, (iii) Solutio: (a) The sequeces {a }ad {b } coverge to the same limit. By R R R & R 3 R 3 λr By R 3 R 3 + R λ λ [A: B] = [ λ ] ~ [ λ λ ] λ λ λ λ λ ~ [ λ λ ] λ λ λ (i) For uique solutio (λ )( λ λ ) (λ ) (λ + ) λ or (ii) (iii) For λ =, we have ifiitely may solutios For λ =, we have o solutio. (b) As a + is arithmetic mea ad b + is harmoic mea betwee a ad b, so a + b +, as A. M. G. M. Hece, we have (i) b a, for all Also b b b 3 mootoically decreasig ad bouded below by b. So it is coverget ad similarly {b } is mootoically icreasig ad bouded above by a. So it is also coverget. Let Lt a = l ad Lt b = m
9 So, {a }ad {b } coverges to same limit. Let l be the commo limit of {a } ad {b } Lt a + = lim a + b l + m l = l = m. a + b + = a b = a b = = a b a b = a b = lim a b = l = l = Hece {a } ad {b } coverges to same limit =. 7. (a) Solve: (x y 3 + xy)dy = dx (b) Fid the geeral solutio of the differetial equatio (D 4D + 4)y = x si x, where D d dx Solutio: (a) (x y 3 + xy)dy = dx dx dy = xy + x y 3 x dx dy = x y + y3 ; let x = z x dx dy = dz dy dz dy = yz + y3 dz + yz = y3 dy ze y = e y ( y 3 )dy + e; Let y = t ydy = dt ze t = e t t dt + c = [tet e t ] + c z = [t ] + ce t x = [y ] + ce y is the solutio. (b) (D 4D + 4)y = x si x (D ) y = x si x Auxiliary equatio is (m ) =
10 m =, So, complemetary fuctio is y = (c + c x)e x Particular itegral is = y = x si x (D ) (D ) = [x (D ) ] si x (D ) = [x D ] D si x 4D + 4 = = [x D ] si x 4D = [x cos x ] D 8 x cos x 8 x cos x 8 = 4 D + D (cos x) 4 + ( si x + cos x) 3 x cos x 8 cos x si x + 6 = [( + x) cos x si x] 6 So, the complete solutio is y = (c + c x)e x + [( + x) cos x si x] 6 8. (a) Fid all the critical poits of the fuctio f(x, y) = x 3 + y 3 + 3xy ad examie those poits for local maxima ad local miima (b) If f is a cotiuous real valued fuctio o [,], show that there exists a poit c (, ) such that Solutio: (a) f(x, y) = x 3 + y 3 + 3xy xf(x)dx = f(x)dx. c
11 f x = 3x + 3y ; f y = 3y + 3x f xx = 6x ; f yy = 6y f xy = 3 For critical poits, we have f x = ad f y = x + y = ad y + x = y = x x 4 + x = x( + x 3 ) = x =, At x =, y = ; at x =, y = So, critical poits are (,) ad (, ) f xx f yy (f xy ) = 36xy 9 = 9(4xy ) At (,); f xx f yy (f xy ) = 9 <. So, (, ) is a saddle poit. At (, ); f xx f yy (f xy ) = 9(4 ) = 7 > Ad f xx at (, ) = 6 <. So(, ) is local maxima poit. (b) xf(x)dx lies betwee f(x)dx ad. f(x)dx Now as f(x) is a cotiuous fuctio i the iterval [,], so f(x)dx f(x)dx Now, as f(x) is cotiuous fuctio, so itegral Hece, f(x)dx exist for each c [,]. c f(x)dx = f(x)dx for a poit c (,) c
12 9. (a) Evaluate the triple itegral z=4 x= z y= 4z x dydxdz z= x= y= (b) Let M = ( ). If M = 5 I + km M, where I is the idetity matrix of order 3, fid the value of k. hece or otherwise, solve the system of equatios: x M ( y) = ( ) z Solutio: (a) z=4 x= z y= 4z x dydxdz = z= x= y= z=4 = z= z=4 x= z 4z x dxdz z= x= [ x 4z x + 4z x= z x si z ] dz = x= z=4 z= πzdz = 8π. so, (b) By Cayley Hamilto theorem, every square matrix satisfies it s characteristic equatio, M λi = is satisfied by M. λ M λi = λ = ( λ)[ + λ 3λ] + λ = λ 3 + 4λ 5λ + 4 Hece, M 3 + 4M 5M + 4I = M + 4M 5I + 4M = (By takig product by M ) M = 4 [M 4M + 5I] = 4 M M I K = 6 Now, M = ( ) M = ( 4 3)
13 M = 4 [M 4M + 5I] = 4 ( ) M = 4 ( ) x x M ( y) = ( ) ( y) = M ( ) z z 4 ( 4 4 x ( y) = z ( ) 4 ) 4 ( 4) x = ; y = 4 & z = 4. (a) Let N be a radom variable represetig the umber of fair dice throw with probability mass fuctio P(N = i) =, i =,,.. i Let S be the sum of umbers appearig o the faces of the dice. Give that S = 3, what is the probability that dice were throw? (b) Let X~ N(, )ad Y = X + X. Fid E(Y 3 ). Solutio: (a) Prob. ( dice were throw sum o the face = 3) = Prob. ( dice were throw sum o the face = 3) Prob. (sum o the face = 3) = P( s = 3) P( s = 3) + P( s = 3) + P(3 s = 3)
14 = = = = (b) X~N(,) Y = X + X = X; X > = ; X Y 3 = 8X 3 X > = ; X E(Y 3 ) = 8x 3. π e x / dx Let z = x dz = dx. Let Y ~N(μ, σ y ) ad Y = l X. E(Y 3 ) = π 6ze z dz = π 6 = 8 π (a) Fid the probability desity fuctio of the radom variable X ad the media of X. (b) Fid the maximum likelihood estimator of the media of the radom variable X based o a radom sample of size. Solutio: (a) Y~N(μ y, σ y ); Y = l X is log ormal distributio whose PDF, f(x) is Media of X = M, the we have f(x) = (l x μ) σ πx e σ ; x > M f(x)dx = M = eμ (b) Likelihood fuctio f L (x μ, σ) = ( x i ) f N (l x; μ, σ) i=
15 f L is pdf of log ormal ad f N is pdf of ormal l L = (μ, σ x, x,, x ) = l x k + l N (μ, σ l x, l x,, l x ) = costat + l N (μ, σ l x, l x,, l x ) So, by usig the cocept of Normal distributio, we have k μ = k l x k ad σ = k (l x k μ ). (a) A radom variable X has probability desity fuctio f(x) = axe β x, x >, α >, β > If E(X) = π, determie α ad β. (b) Let X ad Y be two radom variables with joi probability desity fuctio f(x, y) = { e y if x y otherwise (i) (ii) (iii) Fid the margial desity fuctios of X ad Y Examie whether X ad Y are idepedet Fid Cov (X, Y) Solutio: (a) Let y = (βx) dy = β xdx E(X) = xf(x)dx + αx e β x dx E(X) = α β. y/ β e y dy = α β 3 Γ (3 ) = α π 4β3 π = α = β 3 () Also, f(x)dx = αxe β x dx =
16 α β e y dy = α β = α = β. () From () & () we get β = α = (b) (i) Margial desity fuctio of X, f(x) = e y dy; x < f(x) = e x ; x < Margial desity fuctio of Y, y f(y) = e y dx = ye y = ye y ; y < (ii) As f(x)f(y) = e x (ye y ) = ye (x+y) ad f(x, y) = e y So, f(x, y) f(x)f(y) hece X ad Y are ot idepedet. (iii) E(X) = xe x dx = [ xe x e x ] = E(Y) = y e y dy = [( y y z)e y ] = y E(XY) = xye y dxdy = y3 e y dy y= x= y= = [ y3 3y 6y 6]e y = 3 Cov (X, Y) = E(XY) E(X)E(Y) = 3 ()() = 3 =.
17 3. (a) Let X,., X be a radom sample from Exp ( ) populatio. Obtai the Cramer Rao θ lower boud for the variace of a ubiased estimator of θ. σ. (b) Let X,, X (>4) be a radom sample from a populatio with mea μ ad variace Cosider the followig estimators of μ U = X i, i= V = 8 X 3 + 4( ) (X + + X ) + 8 X (i) (ii) (iii) Examie whether the estimates U ad V are ubiased Examie whether the estimates U ad V are cosistet Which of these two estimates is more efficiet? Justify your aswer. Solutio: (a) If T(X) is ubiased estimator of a fuctio Ψ(θ) of the parameter θ, the Cramer Rao boud of variace of Ψ(θ) is give by Where I(θ) is Fisher s iformatio. V(T(X)) [Ψ (θ)] L(x, θ) I(θ) = E [( ) θ I(θ) L(x, θ) = log f(x, θ) ] = E [ L(x; θ) θ ] Give Ψ(θ) = θ Ψ (θ) = θ f(x i= θ) = θ e x/θ ; θ >, x > f(x, θ) = θ e xi/θ = θ i= e xi/θ L(x, θ) = log f(x, θ) = log θ ( xi)/θ e ( xi) = log θ θ = I(θ) = E [( θ + x i θ ) ]
18 Var [T(x)] 4θ E [( θ + xi θ ) ] So, lower boud for variace is 4θ E [( θ + xi θ ) ] (b) U = X i i= V = 8 X 3 + 4( ) (X + + X ) + 8 X (i) E(U) = E(X i) = μ = μ i= E(X) = μ ( ) 4 ( ) μ + 8 μ = μ ( ) = μ Both U ad V are ubiased estimator of μ as E(U) = E(V) = μ (ii) lim U = μ ad lim V = μ So, both U ad V are cosistet estimator of μ. (iii) As, Var(U) < Var (V) Var(U) = Var ( X i) = [Var X + Var X + + Var X ] i= = [σ + σ + + σ ] = σ Var (V) = 9 64 σ + 6( ) ( )σ + 64 σ = 9 3 σ + 6( ) σ
19 Here U is more efficiet estimator tha V. 4. Let X,, X be a radom sample from a Beroulli populatio with parameter p, (a) (i) Fid a sufficiet statistic for p (ii) Cosider a estimator U(X, X ) of P( p) give by U(X, X ) = { if X + X = otherwise Examie whether U(X, X ) is the a ubiased estimator (b) Usig the results obtaied i (a) above ad Rao Blackwell theorem, fid the uiformity miimum variace ubiased estimator (UMVUE) of Solutio: (a) p( p). (i) T(X) = X + X + + X is sufficiet statistics for p. As P r = (X = x) = Pr{X = x, X = x,., X = x } = P x ( p) x p x ( p) x P x ( p) x = P (x +x + +x )( p) (x +x+ +x ) = p x i( p) x i = P T(X) ( P) T(X) Hece T(x) = x i (ii) X + X = Evaluate U(X, X ) ad if E(U(X, X )) = P( p) the U is ubiased E(U) = (X =, X = ) + (X =, X = ) = p( p) p( P) p( p) + = Hece U is a ubiased estimator of p( p). (b) Now by Rao Blackwell theorem, we obtai UMVUE, which requires the statistics to be sufficiet.
20 Also, mea square error of Rao Blackwell estimator does ot exceed that of the origial estimator. Rest of the calculatios are left for you as a exercise. 5. (a) Let X,, X be a radom sample from the populatio havig probability desity fuctio x f(x, θ) = { e θ θ if x > otherwise Obtai the most powerful test for testig H : θ = θ agaist H : θ = θ (θ < θ ) x (b) Let X,, X be a radom sample of size from N(μ, ) populatio. To test H μ = 5 agaist H : μ = 4, the decisio rule is : Reject H if x c. If α =. 5 ad β =., determie (rouded off to a iteger) ad hece c. Solutio: (a) Likelihood fuctio, L(x, θ) = x i θ e x i /θ i= Cosider H : θ = θ & H : θ = θ The best critical regio, usig Neyma Pearso Lemma is give by = θ x ie x i /θi k i= θ i= x i e x i /θ K ( θ ) e x i ( θ θ ) θ i= log K (log θ log θ ) + ( θ θ ) x i Now P ( ( θ θ ) x i log k (log θ log θ ) H is true) = α i= So it is used for most powerful test. (b) Same as above. Do yourself. i=
IIT JAM Mathematical Statistics (MS) 2006 SECTION A
IIT JAM Mathematical Statistics (MS) 6 SECTION A. If a > for ad lim a / L >, the which of the followig series is ot coverget? (a) (b) (c) (d) (d) = = a = a = a a + / a lim a a / + = lim a / a / + = lim
More informationMathematical Statistics - MS
Paper Specific Istructios. The examiatio is of hours duratio. There are a total of 60 questios carryig 00 marks. The etire paper is divided ito three sectios, A, B ad C. All sectios are compulsory. Questios
More informationUnbiased Estimation. February 7-12, 2008
Ubiased Estimatio February 7-2, 2008 We begi with a sample X = (X,..., X ) of radom variables chose accordig to oe of a family of probabilities P θ where θ is elemet from the parameter space Θ. For radom
More informationDirection: This test is worth 250 points. You are required to complete this test within 50 minutes.
Term Test October 3, 003 Name Math 56 Studet Number Directio: This test is worth 50 poits. You are required to complete this test withi 50 miutes. I order to receive full credit, aswer each problem completely
More informationAssignment 1 : Real Numbers, Sequences. for n 1. Show that (x n ) converges. Further, by observing that x n+2 + x n+1
Assigmet : Real Numbers, Sequeces. Let A be a o-empty subset of R ad α R. Show that α = supa if ad oly if α is ot a upper boud of A but α + is a upper boud of A for every N. 2. Let y (, ) ad x (, ). Evaluate
More informationAsymptotics. Hypothesis Testing UMP. Asymptotic Tests and p-values
of the secod half Biostatistics 6 - Statistical Iferece Lecture 6 Fial Exam & Practice Problems for the Fial Hyu Mi Kag Apil 3rd, 3 Hyu Mi Kag Biostatistics 6 - Lecture 6 Apil 3rd, 3 / 3 Rao-Blackwell
More informationThis exam contains 19 pages (including this cover page) and 10 questions. A Formulae sheet is provided with the exam.
Probability ad Statistics FS 07 Secod Sessio Exam 09.0.08 Time Limit: 80 Miutes Name: Studet ID: This exam cotais 9 pages (icludig this cover page) ad 0 questios. A Formulae sheet is provided with the
More informationEE 4TM4: Digital Communications II Probability Theory
1 EE 4TM4: Digital Commuicatios II Probability Theory I. RANDOM VARIABLES A radom variable is a real-valued fuctio defied o the sample space. Example: Suppose that our experimet cosists of tossig two fair
More informationRandom Variables, Sampling and Estimation
Chapter 1 Radom Variables, Samplig ad Estimatio 1.1 Itroductio This chapter will cover the most importat basic statistical theory you eed i order to uderstad the ecoometric material that will be comig
More informationChapter 6 Infinite Series
Chapter 6 Ifiite Series I the previous chapter we cosidered itegrals which were improper i the sese that the iterval of itegratio was ubouded. I this chapter we are goig to discuss a topic which is somewhat
More informationEECS564 Estimation, Filtering, and Detection Hwk 2 Solns. Winter p θ (z) = (2θz + 1 θ), 0 z 1
EECS564 Estimatio, Filterig, ad Detectio Hwk 2 Sols. Witer 25 4. Let Z be a sigle observatio havig desity fuctio where. p (z) = (2z + ), z (a) Assumig that is a oradom parameter, fid ad plot the maximum
More informationStatistical Theory MT 2009 Problems 1: Solution sketches
Statistical Theory MT 009 Problems : Solutio sketches. Which of the followig desities are withi a expoetial family? Explai your reasoig. (a) Let 0 < θ < ad put f(x, θ) = ( θ)θ x ; x = 0,,,... (b) (c) where
More informationMath 132, Fall 2009 Exam 2: Solutions
Math 3, Fall 009 Exam : Solutios () a) ( poits) Determie for which positive real umbers p, is the followig improper itegral coverget, ad for which it is diverget. Evaluate the itegral for each value of
More informationStatistical Theory MT 2008 Problems 1: Solution sketches
Statistical Theory MT 008 Problems : Solutio sketches. Which of the followig desities are withi a expoetial family? Explai your reasoig. a) Let 0 < θ < ad put fx, θ) = θ)θ x ; x = 0,,,... b) c) where α
More informationSolutions: Homework 3
Solutios: Homework 3 Suppose that the radom variables Y,...,Y satisfy Y i = x i + " i : i =,..., IID where x,...,x R are fixed values ad ",...," Normal(0, )with R + kow. Fid ˆ = MLE( ). IND Solutio: Observe
More informationProblem Set 4 Due Oct, 12
EE226: Radom Processes i Systems Lecturer: Jea C. Walrad Problem Set 4 Due Oct, 12 Fall 06 GSI: Assae Gueye This problem set essetially reviews detectio theory ad hypothesis testig ad some basic otios
More informationMATHEMATICAL SCIENCES PAPER-II
MATHEMATICAL SCIENCES PAPER-II. Let {x } ad {y } be two sequeces of real umbers. Prove or disprove each of the statemets :. If {x y } coverges, ad if {y } is coverget, the {x } is coverget.. {x + y } coverges
More informationFourier Series and their Applications
Fourier Series ad their Applicatios The fuctios, cos x, si x, cos x, si x, are orthogoal over (, ). m cos mx cos xdx = m = m = = cos mx si xdx = for all m, { m si mx si xdx = m = I fact the fuctios satisfy
More informationFirst Year Quantitative Comp Exam Spring, Part I - 203A. f X (x) = 0 otherwise
First Year Quatitative Comp Exam Sprig, 2012 Istructio: There are three parts. Aswer every questio i every part. Questio I-1 Part I - 203A A radom variable X is distributed with the margial desity: >
More informationMH1101 AY1617 Sem 2. Question 1. NOT TESTED THIS TIME
MH AY67 Sem Questio. NOT TESTED THIS TIME ( marks Let R be the regio bouded by the curve y 4x x 3 ad the x axis i the first quadrat (see figure below. Usig the cylidrical shell method, fid the volume of
More informationStat 319 Theory of Statistics (2) Exercises
Kig Saud Uiversity College of Sciece Statistics ad Operatios Research Departmet Stat 39 Theory of Statistics () Exercises Refereces:. Itroductio to Mathematical Statistics, Sixth Editio, by R. Hogg, J.
More informationPAPER : IIT-JAM 2010
MATHEMATICS-MA (CODE A) Q.-Q.5: Oly oe optio is correct for each questio. Each questio carries (+6) marks for correct aswer ad ( ) marks for icorrect aswer.. Which of the followig coditios does NOT esure
More information62. Power series Definition 16. (Power series) Given a sequence {c n }, the series. c n x n = c 0 + c 1 x + c 2 x 2 + c 3 x 3 +
62. Power series Defiitio 16. (Power series) Give a sequece {c }, the series c x = c 0 + c 1 x + c 2 x 2 + c 3 x 3 + is called a power series i the variable x. The umbers c are called the coefficiets of
More informationWe are mainly going to be concerned with power series in x, such as. (x)} converges - that is, lims N n
Review of Power Series, Power Series Solutios A power series i x - a is a ifiite series of the form c (x a) =c +c (x a)+(x a) +... We also call this a power series cetered at a. Ex. (x+) is cetered at
More informationStatistical Inference (Chapter 10) Statistical inference = learn about a population based on the information provided by a sample.
Statistical Iferece (Chapter 10) Statistical iferece = lear about a populatio based o the iformatio provided by a sample. Populatio: The set of all values of a radom variable X of iterest. Characterized
More informationProbability 2 - Notes 10. Lemma. If X is a random variable and g(x) 0 for all x in the support of f X, then P(g(X) 1) E[g(X)].
Probability 2 - Notes 0 Some Useful Iequalities. Lemma. If X is a radom variable ad g(x 0 for all x i the support of f X, the P(g(X E[g(X]. Proof. (cotiuous case P(g(X Corollaries x:g(x f X (xdx x:g(x
More informationThis section is optional.
4 Momet Geeratig Fuctios* This sectio is optioal. The momet geeratig fuctio g : R R of a radom variable X is defied as g(t) = E[e tx ]. Propositio 1. We have g () (0) = E[X ] for = 1, 2,... Proof. Therefore
More informationLecture 15: Density estimation
Lecture 15: Desity estimatio Why do we estimate a desity? Suppose that X 1,...,X are i.i.d. radom variables from F ad that F is ukow but has a Lebesgue p.d.f. f. Estimatio of F ca be doe by estimatig f.
More informationMATH 320: Probability and Statistics 9. Estimation and Testing of Parameters. Readings: Pruim, Chapter 4
MATH 30: Probability ad Statistics 9. Estimatio ad Testig of Parameters Estimatio ad Testig of Parameters We have bee dealig situatios i which we have full kowledge of the distributio of a radom variable.
More informationReview for Test 3 Math 1552, Integral Calculus Sections 8.8,
Review for Test 3 Math 55, Itegral Calculus Sectios 8.8, 0.-0.5. Termiology review: complete the followig statemets. (a) A geometric series has the geeral form k=0 rk.theseriescovergeswhe r is less tha
More informationEcon 325 Notes on Point Estimator and Confidence Interval 1 By Hiro Kasahara
Poit Estimator Eco 325 Notes o Poit Estimator ad Cofidece Iterval 1 By Hiro Kasahara Parameter, Estimator, ad Estimate The ormal probability desity fuctio is fully characterized by two costats: populatio
More informationSome Basic Probability Concepts. 2.1 Experiments, Outcomes and Random Variables
Some Basic Probability Cocepts 2. Experimets, Outcomes ad Radom Variables A radom variable is a variable whose value is ukow util it is observed. The value of a radom variable results from a experimet;
More information6. Sufficient, Complete, and Ancillary Statistics
Sufficiet, Complete ad Acillary Statistics http://www.math.uah.edu/stat/poit/sufficiet.xhtml 1 of 7 7/16/2009 6:13 AM Virtual Laboratories > 7. Poit Estimatio > 1 2 3 4 5 6 6. Sufficiet, Complete, ad Acillary
More informationFINALTERM EXAMINATION Fall 9 Calculus & Aalytical Geometry-I Questio No: ( Mars: ) - Please choose oe Let f ( x) is a fuctio such that as x approaches a real umber a, either from left or right-had-side,
More informationLecture 19: Convergence
Lecture 19: Covergece Asymptotic approach I statistical aalysis or iferece, a key to the success of fidig a good procedure is beig able to fid some momets ad/or distributios of various statistics. I may
More informationSequences and Series of Functions
Chapter 6 Sequeces ad Series of Fuctios 6.1. Covergece of a Sequece of Fuctios Poitwise Covergece. Defiitio 6.1. Let, for each N, fuctio f : A R be defied. If, for each x A, the sequece (f (x)) coverges
More informationMath 142, Final Exam. 5/2/11.
Math 4, Fial Exam 5// No otes, calculator, or text There are poits total Partial credit may be give Write your full ame i the upper right corer of page Number the pages i the upper right corer Do problem
More informationMATH 31B: MIDTERM 2 REVIEW
MATH 3B: MIDTERM REVIEW JOE HUGHES. Evaluate x (x ) (x 3).. Partial Fractios Solutio: The umerator has degree less tha the deomiator, so we ca use partial fractios. Write x (x ) (x 3) = A x + A (x ) +
More informationQuick Review of Probability
Quick Review of Probability Berli Che Departmet of Computer Sciece & Iformatio Egieerig Natioal Taiwa Normal Uiversity Refereces: 1. W. Navidi. Statistics for Egieerig ad Scietists. Chapter & Teachig Material.
More informationIndian Institute of Information Technology, Allahabad. End Semester Examination - Tentative Marking Scheme
Idia Istitute of Iformatio Techology, Allahabad Ed Semester Examiatio - Tetative Markig Scheme Course Name: Mathematics-I Course Code: SMAT3C MM: 75 Program: B.Tech st year (IT+ECE) ate of Exam:..7 ( st
More informationSTAT Homework 1 - Solutions
STAT-36700 Homework 1 - Solutios Fall 018 September 11, 018 This cotais solutios for Homework 1. Please ote that we have icluded several additioal commets ad approaches to the problems to give you better
More informationMTH Assignment 1 : Real Numbers, Sequences
MTH -26 Assigmet : Real Numbers, Sequeces. Fid the supremum of the set { m m+ : N, m Z}. 2. Let A be a o-empty subset of R ad α R. Show that α = supa if ad oly if α is ot a upper boud of A but α + is a
More information1 Lecture 2: Sequence, Series and power series (8/14/2012)
Summer Jump-Start Program for Aalysis, 202 Sog-Yig Li Lecture 2: Sequece, Series ad power series (8/4/202). More o sequeces Example.. Let {x } ad {y } be two bouded sequeces. Show lim sup (x + y ) lim
More informationSOLUTIONS TO EXAM 3. Solution: Note that this defines two convergent geometric series with respective radii r 1 = 2/5 < 1 and r 2 = 1/5 < 1.
SOLUTIONS TO EXAM 3 Problem Fid the sum of the followig series 2 + ( ) 5 5 2 5 3 25 2 2 This series diverges Solutio: Note that this defies two coverget geometric series with respective radii r 2/5 < ad
More information17. Joint distributions of extreme order statistics Lehmann 5.1; Ferguson 15
17. Joit distributios of extreme order statistics Lehma 5.1; Ferguso 15 I Example 10., we derived the asymptotic distributio of the maximum from a radom sample from a uiform distributio. We did this usig
More informationFACULTY OF MATHEMATICAL STUDIES MATHEMATICS FOR PART I ENGINEERING. Lectures
FACULTY OF MATHEMATICAL STUDIES MATHEMATICS FOR PART I ENGINEERING Lectures MODULE 5 STATISTICS II. Mea ad stadard error of sample data. Biomial distributio. Normal distributio 4. Samplig 5. Cofidece itervals
More informationJanuary 25, 2017 INTRODUCTION TO MATHEMATICAL STATISTICS
Jauary 25, 207 INTRODUCTION TO MATHEMATICAL STATISTICS Abstract. A basic itroductio to statistics assumig kowledge of probability theory.. Probability I a typical udergraduate problem i probability, we
More informationLet us give one more example of MLE. Example 3. The uniform distribution U[0, θ] on the interval [0, θ] has p.d.f.
Lecture 5 Let us give oe more example of MLE. Example 3. The uiform distributio U[0, ] o the iterval [0, ] has p.d.f. { 1 f(x =, 0 x, 0, otherwise The likelihood fuctio ϕ( = f(x i = 1 I(X 1,..., X [0,
More informationChapter 6 Principles of Data Reduction
Chapter 6 for BST 695: Special Topics i Statistical Theory. Kui Zhag, 0 Chapter 6 Priciples of Data Reductio Sectio 6. Itroductio Goal: To summarize or reduce the data X, X,, X to get iformatio about a
More information2.4.2 A Theorem About Absolutely Convergent Series
0 Versio of August 27, 200 CHAPTER 2. INFINITE SERIES Add these two series: + 3 2 + 5 + 7 4 + 9 + 6 +... = 3 l 2. (2.20) 2 Sice the reciprocal of each iteger occurs exactly oce i the last series, we would
More informationChapter 2 The Monte Carlo Method
Chapter 2 The Mote Carlo Method The Mote Carlo Method stads for a broad class of computatioal algorithms that rely o radom sampligs. It is ofte used i physical ad mathematical problems ad is most useful
More informationTopic 5 [434 marks] (i) Find the range of values of n for which. (ii) Write down the value of x dx in terms of n, when it does exist.
Topic 5 [44 marks] 1a (i) Fid the rage of values of for which eists 1 Write dow the value of i terms of 1, whe it does eist Fid the solutio to the differetial equatio 1b give that y = 1 whe = π (cos si
More informationn 3 ln n n ln n is convergent by p-series for p = 2 > 1. n2 Therefore we can apply Limit Comparison Test to determine lutely convergent.
06 微甲 0-04 06-0 班期中考解答和評分標準. ( poits) Determie whether the series is absolutely coverget, coditioally coverget, or diverget. Please state the tests which you use. (a) ( poits) (b) ( poits) (c) ( poits)
More informationLecture 18: Sampling distributions
Lecture 18: Samplig distributios I may applicatios, the populatio is oe or several ormal distributios (or approximately). We ow study properties of some importat statistics based o a radom sample from
More informationDistribution of Random Samples & Limit theorems
STAT/MATH 395 A - PROBABILITY II UW Witer Quarter 2017 Néhémy Lim Distributio of Radom Samples & Limit theorems 1 Distributio of i.i.d. Samples Motivatig example. Assume that the goal of a study is to
More informationStat410 Probability and Statistics II (F16)
Some Basic Cocepts of Statistical Iferece (Sec 5.) Suppose we have a rv X that has a pdf/pmf deoted by f(x; θ) or p(x; θ), where θ is called the parameter. I previous lectures, we focus o probability problems
More informationECONOMETRIC THEORY. MODULE XIII Lecture - 34 Asymptotic Theory and Stochastic Regressors
ECONOMETRIC THEORY MODULE XIII Lecture - 34 Asymptotic Theory ad Stochastic Regressors Dr. Shalabh Departmet of Mathematics ad Statistics Idia Istitute of Techology Kapur Asymptotic theory The asymptotic
More informationMath 12 Final Exam, May 11, 2011 ANSWER KEY. 2sinh(2x) = lim. 1 x. lim e. x ln. = e. (x+1)(1) x(1) (x+1) 2. (2secθ) 5 2sec2 θ dθ.
Math Fial Exam, May, ANSWER KEY. [5 Poits] Evaluate each of the followig its. Please justify your aswers. Be clear if the it equals a value, + or, or Does Not Exist. coshx) a) L H x x+l x) sihx) x x L
More informationThe variance of a sum of independent variables is the sum of their variances, since covariances are zero. Therefore. V (xi )= n n 2 σ2 = σ2.
SAMPLE STATISTICS A radom sample x 1,x,,x from a distributio f(x) is a set of idepedetly ad idetically variables with x i f(x) for all i Their joit pdf is f(x 1,x,,x )=f(x 1 )f(x ) f(x )= f(x i ) The sample
More informationd) If the sequence of partial sums converges to a limit L, we say that the series converges and its
Ifiite Series. Defiitios & covergece Defiitio... Let {a } be a sequece of real umbers. a) A expressio of the form a + a +... + a +... is called a ifiite series. b) The umber a is called as the th term
More informationDiscrete Mathematics for CS Spring 2008 David Wagner Note 22
CS 70 Discrete Mathematics for CS Sprig 2008 David Wager Note 22 I.I.D. Radom Variables Estimatig the bias of a coi Questio: We wat to estimate the proportio p of Democrats i the US populatio, by takig
More informationSTAT Homework 2 - Solutions
STAT-36700 Homework - Solutios Fall 08 September 4, 08 This cotais solutios for Homework. Please ote that we have icluded several additioal commets ad approaches to the problems to give you better isight.
More informationHOMEWORK I: PREREQUISITES FROM MATH 727
HOMEWORK I: PREREQUISITES FROM MATH 727 Questio. Let X, X 2,... be idepedet expoetial radom variables with mea µ. (a) Show that for Z +, we have EX µ!. (b) Show that almost surely, X + + X (c) Fid the
More informationQuick Review of Probability
Quick Review of Probability Berli Che Departmet of Computer Sciece & Iformatio Egieerig Natioal Taiwa Normal Uiversity Refereces: 1. W. Navidi. Statistics for Egieerig ad Scietists. Chapter 2 & Teachig
More informationMA541 : Real Analysis. Tutorial and Practice Problems - 1 Hints and Solutions
MA54 : Real Aalysis Tutorial ad Practice Problems - Hits ad Solutios. Suppose that S is a oempty subset of real umbers that is bouded (i.e. bouded above as well as below). Prove that if S sup S. What ca
More informationApproximations and more PMFs and PDFs
Approximatios ad more PMFs ad PDFs Saad Meimeh 1 Approximatio of biomial with Poisso Cosider the biomial distributio ( b(k,,p = p k (1 p k, k λ: k Assume that is large, ad p is small, but p λ at the limit.
More informationB U Department of Mathematics Math 101 Calculus I
B U Departmet of Mathematics Math Calculus I Sprig 5 Fial Exam Calculus archive is a property of Boğaziçi Uiversity Mathematics Departmet. The purpose of this archive is to orgaise ad cetralise the distributio
More informationProbability and Statistics
ICME Refresher Course: robability ad Statistics Staford Uiversity robability ad Statistics Luyag Che September 20, 2016 1 Basic robability Theory 11 robability Spaces A probability space is a triple (Ω,
More informationExpectation and Variance of a random variable
Chapter 11 Expectatio ad Variace of a radom variable The aim of this lecture is to defie ad itroduce mathematical Expectatio ad variace of a fuctio of discrete & cotiuous radom variables ad the distributio
More informationJANE PROFESSOR WW Prob Lib1 Summer 2000
JANE PROFESSOR WW Prob Lib Summer 000 Sample WeBWorK problems. WeBWorK assigmet Series6CompTests due /6/06 at :00 AM..( pt) Test each of the followig series for covergece by either the Compariso Test or
More informationMIDTERM 3 CALCULUS 2. Monday, December 3, :15 PM to 6:45 PM. Name PRACTICE EXAM SOLUTIONS
MIDTERM 3 CALCULUS MATH 300 FALL 08 Moday, December 3, 08 5:5 PM to 6:45 PM Name PRACTICE EXAM S Please aswer all of the questios, ad show your work. You must explai your aswers to get credit. You will
More informationMath 210A Homework 1
Math 0A Homework Edward Burkard Exercise. a) State the defiitio of a aalytic fuctio. b) What are the relatioships betwee aalytic fuctios ad the Cauchy-Riema equatios? Solutio. a) A fuctio f : G C is called
More informationEstimation for Complete Data
Estimatio for Complete Data complete data: there is o loss of iformatio durig study. complete idividual complete data= grouped data A complete idividual data is the oe i which the complete iformatio of
More informationST5215: Advanced Statistical Theory
ST525: Advaced Statistical Theory Departmet of Statistics & Applied Probability Tuesday, September 7, 2 ST525: Advaced Statistical Theory Lecture : The law of large umbers The Law of Large Numbers The
More informationLast Lecture. Wald Test
Last Lecture Biostatistics 602 - Statistical Iferece Lecture 22 Hyu Mi Kag April 9th, 2013 Is the exact distributio of LRT statistic typically easy to obtai? How about its asymptotic distributio? For testig
More informationSUMMARY OF SEQUENCES AND SERIES
SUMMARY OF SEQUENCES AND SERIES Importat Defiitios, Results ad Theorems for Sequeces ad Series Defiitio. A sequece {a } has a limit L ad we write lim a = L if for every ɛ > 0, there is a correspodig iteger
More informationDirection: This test is worth 150 points. You are required to complete this test within 55 minutes.
Term Test 3 (Part A) November 1, 004 Name Math 6 Studet Number Directio: This test is worth 10 poits. You are required to complete this test withi miutes. I order to receive full credit, aswer each problem
More informationMath 10A final exam, December 16, 2016
Please put away all books, calculators, cell phoes ad other devices. You may cosult a sigle two-sided sheet of otes. Please write carefully ad clearly, USING WORDS (ot just symbols). Remember that the
More informationIn this section, we show how to use the integral test to decide whether a series
Itegral Test Itegral Test Example Itegral Test Example p-series Compariso Test Example Example 2 Example 3 Example 4 Example 5 Exa Itegral Test I this sectio, we show how to use the itegral test to decide
More informationMidterm Exam #2. Please staple this cover and honor pledge atop your solutions.
Math 50B Itegral Calculus April, 07 Midterm Exam # Name: Aswer Key David Arold Istructios. (00 poits) This exam is ope otes, ope book. This icludes ay supplemetary texts or olie documets. You are ot allowed
More informationMATH 472 / SPRING 2013 ASSIGNMENT 2: DUE FEBRUARY 4 FINALIZED
MATH 47 / SPRING 013 ASSIGNMENT : DUE FEBRUARY 4 FINALIZED Please iclude a cover sheet that provides a complete setece aswer to each the followig three questios: (a) I your opiio, what were the mai ideas
More informationMathematics 170B Selected HW Solutions.
Mathematics 17B Selected HW Solutios. F 4. Suppose X is B(,p). (a)fidthemometgeeratigfuctiom (s)of(x p)/ p(1 p). Write q = 1 p. The MGF of X is (pe s + q), sice X ca be writte as the sum of idepedet Beroulli
More informationLecture 7: Properties of Random Samples
Lecture 7: Properties of Radom Samples 1 Cotiued From Last Class Theorem 1.1. Let X 1, X,...X be a radom sample from a populatio with mea µ ad variace σ
More informationLecture 11 and 12: Basic estimation theory
Lecture ad 2: Basic estimatio theory Sprig 202 - EE 94 Networked estimatio ad cotrol Prof. Kha March 2 202 I. MAXIMUM-LIKELIHOOD ESTIMATORS The maximum likelihood priciple is deceptively simple. Louis
More informationLecture 6 Simple alternatives and the Neyman-Pearson lemma
STATS 00: Itroductio to Statistical Iferece Autum 06 Lecture 6 Simple alteratives ad the Neyma-Pearso lemma Last lecture, we discussed a umber of ways to costruct test statistics for testig a simple ull
More informationMath 113, Calculus II Winter 2007 Final Exam Solutions
Math, Calculus II Witer 7 Fial Exam Solutios (5 poits) Use the limit defiitio of the defiite itegral ad the sum formulas to compute x x + dx The check your aswer usig the Evaluatio Theorem Solutio: I this
More informationIntroduction to Extreme Value Theory Laurens de Haan, ISM Japan, Erasmus University Rotterdam, NL University of Lisbon, PT
Itroductio to Extreme Value Theory Laures de Haa, ISM Japa, 202 Itroductio to Extreme Value Theory Laures de Haa Erasmus Uiversity Rotterdam, NL Uiversity of Lisbo, PT Itroductio to Extreme Value Theory
More informationINFINITE SEQUENCES AND SERIES
11 INFINITE SEQUENCES AND SERIES INFINITE SEQUENCES AND SERIES 11.4 The Compariso Tests I this sectio, we will lear: How to fid the value of a series by comparig it with a kow series. COMPARISON TESTS
More informationParameter, Statistic and Random Samples
Parameter, Statistic ad Radom Samples A parameter is a umber that describes the populatio. It is a fixed umber, but i practice we do ot kow its value. A statistic is a fuctio of the sample data, i.e.,
More information5. Likelihood Ratio Tests
1 of 5 7/29/2009 3:16 PM Virtual Laboratories > 9. Hy pothesis Testig > 1 2 3 4 5 6 7 5. Likelihood Ratio Tests Prelimiaries As usual, our startig poit is a radom experimet with a uderlyig sample space,
More informationCHAPTER 10 INFINITE SEQUENCES AND SERIES
CHAPTER 10 INFINITE SEQUENCES AND SERIES 10.1 Sequeces 10.2 Ifiite Series 10.3 The Itegral Tests 10.4 Compariso Tests 10.5 The Ratio ad Root Tests 10.6 Alteratig Series: Absolute ad Coditioal Covergece
More informationf(x) dx as we do. 2x dx x also diverges. Solution: We compute 2x dx lim
Math 3, Sectio 2. (25 poits) Why we defie f(x) dx as we do. (a) Show that the improper itegral diverges. Hece the improper itegral x 2 + x 2 + b also diverges. Solutio: We compute x 2 + = lim b x 2 + =
More informationMathematical Methods for Physics and Engineering
Mathematical Methods for Physics ad Egieerig Lecture otes Sergei V. Shabaov Departmet of Mathematics, Uiversity of Florida, Gaiesville, FL 326 USA CHAPTER The theory of covergece. Numerical sequeces..
More informationTAMS24: Notations and Formulas
TAMS4: Notatios ad Formulas Basic otatios ad defiitios X: radom variable stokastiska variabel Mea Vätevärde: µ = X = by Xiagfeg Yag kpx k, if X is discrete, xf Xxdx, if X is cotiuous Variace Varias: =
More informationINFINITE SEQUENCES AND SERIES
INFINITE SEQUENCES AND SERIES INFINITE SEQUENCES AND SERIES I geeral, it is difficult to fid the exact sum of a series. We were able to accomplish this for geometric series ad the series /[(+)]. This is
More informationNotes 5 : More on the a.s. convergence of sums
Notes 5 : More o the a.s. covergece of sums Math 733-734: Theory of Probability Lecturer: Sebastie Roch Refereces: Dur0, Sectios.5; Wil9, Sectio 4.7, Shi96, Sectio IV.4, Dur0, Sectio.. Radom series. Three-series
More informationCalculus 2 - D. Yuen Final Exam Review (Version 11/22/2017. Please report any possible typos.)
Calculus - D Yue Fial Eam Review (Versio //7 Please report ay possible typos) NOTE: The review otes are oly o topics ot covered o previous eams See previous review sheets for summary of previous topics
More informationSince X n /n P p, we know that X n (n. Xn (n X n ) Using the asymptotic result above to obtain an approximation for fixed n, we obtain
Assigmet 9 Exercise 5.5 Let X biomial, p, where p 0, 1 is ukow. Obtai cofidece itervals for p i two differet ways: a Sice X / p d N0, p1 p], the variace of the limitig distributio depeds oly o p. Use the
More informationIntroductory statistics
CM9S: Machie Learig for Bioiformatics Lecture - 03/3/06 Itroductory statistics Lecturer: Sriram Sakararama Scribe: Sriram Sakararama We will provide a overview of statistical iferece focussig o the key
More informationINTRODUCTORY MATHEMATICAL ANALYSIS
INTRODUCTORY MATHEMATICAL ANALYSIS For Busiess, Ecoomics, ad the Life ad Social Scieces Chapter 4 Itegratio 0 Pearso Educatio, Ic. Chapter 4: Itegratio Chapter Objectives To defie the differetial. To defie
More information