Plane Answers To Complex Questions 4th Edition Ronald Christensen - Solutions

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Exercise 12.5 Show that d is approximately equal to 2(1 − ra), where ra is the sample (auto)correlation between the pairs ( ˆei+1, ˆei ) i = 1, . . . , n − 1.
Exercise 12.4 Using the model of Example 12.2.2, estimate the power of detecting a t (3) with α = 0.05 by simulation.
Exercise 12.3 Show that the sample correlation between two (mean adjusted)vectors in the same direction is 1.
Exercise 12.2 Showthat for a regression model that does not contain an intercept, the diagonal elements of the perpendicular projection operator are equivalent to the estimated squared Mahalanobis distances computed with the assumption that μ = 0.
Exercise 12.1 Show that for a standard linear model ˆe is the BLUP of e.
Exercise 11.10 Consider the table of meansτ1 2 · · · m 1 ¯y·11 ¯y·12 · · · ¯y·1mω 2 ¯y·21 ¯y·22 · · · ¯y·2m.......... . ....t ¯y·t1 ¯y·t2 · · · ¯y·tm
Exercise 11.9 Show that model (11.1.5) for a randomized complete block design is a generalized split plot model.
Exercise 11.8 Give detailed proofs of the test statistic’s distribution for(a) testing model (8) against model (2),(b) testing model (9) against model (2).
Exercise 11.7 Verify that Condition 11.2.3 holds for model (3).
Exercise 11.6 Show that when usingmodel (5) for a randomized complete block design, the BLUE of a contrast in the αi s is the same regardless of whether the βj s are assumed random or fixed. Show that the estimates of a contrast’s variance are the same.
Exercise 11.5 An alternative model for a block design is yi j = μ + αi + βj + ei j , (5)where the βj s are independent N(0, σ2 2 ) and the βj s and ei j s are independent. If this model is used for a balanced incomplete block design, the BLUEs are different than they are when the βj s are
Exercise 11.4 The usual model for a randomized complete block design was given in Section 8.2 as yi j = μ + αi + βj + ei j , i = 1, . . . ,a, j = 1, . . . ,b, Var(ei j ) = σ2, and Cov(ei j , ei j ) = 0 for (i, j ) =(i , j ). The βj s are considered as fixed block effects. Consider now a
Exercise 11.3 Prove Theorem 11.1.3(i).
Exercise 11.2 Prove Theorem 11.1.1.Hint: For (i), use Theorem 1.3.6. For (ii), show that Cov[(I − M)Y, MY] = 0.
Exercise 11.1 Prove that equation (4) is true.
Exercise 10.15 Let C(V0) = C(X) ∩ C(V), C(X) = C(V0, X1), C(V) =C(V0, V1) with the columns of V0, V1, and X1 being orthonormal. Show that the columns of [V0, V1, X1] are linearly independent.Hint: Write V0b0 + V1b1 + X1b2 = 0 and show that bi = 0, i = 0, 1, 2. In particular, write 0.5V0b0 + V1b1
Exercise 10.14 Show that the results in this section do not depend on the particular choice of Q.
Exercise 10.13 Show that Cov(BY ) = 0 iff B = B∗ Q for some matrix B∗.Hint: First, decompose B into the sum of two matrices B0 and B1 with C(B0) ⊂C(V) and C(B1) ⊥ C(V). Then use a singular value decomposition of V to show that B0V B0 = 0 iff B0 = 0.
Exercise 10.11 Use Definition B.31 and Proposition 10.4.6 to show that MC(X)∩C(V) = M − MW where C(W) = C[M(I − MV )].Another way to find V0 is due to Rao and Mitra (1971, pp. 118, 119). V0 =X ˜U where C( ˜U ) = C[X(I − MV )]⊥. Note that (I − MV )X ˜U = 0, so C(X ˜U ) ⊂C(V0). To
Exercise 10.10 Show that ordinary least squares estimates are best linear unbiased estimates in the model Y = Xβ +e, E(e) = 0, Cov(e) = V if the columns of X are eigenvectors of V.
Exercise 10.9 From inspecting Figure 10.2, give the least squares CLUE for Example 10.2.2. Do not do any matrix manipulations.Remark Suppose that we are analyzing the model Y = Xβ +e, E(e) = 0, Cov(e) = Σ(θ), where Σ(θ) is some nonnegative definite matrix depending on a vector of unknown
Exercise 10.8 Give the general form for a BLUE of Xβ in model (10.1.1).Hint: Add something to a particular BLUE.
Exercise 10.7 Show that if MY − MV1γˆ is a BLUE of Xβ, then C(VV0) ⊂C(V0).Hint: Multiply on the right by V0 after showing thatM − MV1 V1(I − M)V1−1 V1(I − M)V= VM − (I − M)V1 V1(I − M)V1−1 V1M.We include a result that allows one to find thematrix V1, and thus find
Exercise 10.6 Show that V1(I − M)V1 is invertible.Answer: The columns of V1 form a basis, so 0 = V1b iff b = 0. Also(I − M)V1b = 0 iff V1b ∈ C(X), but V1b ∈ C(X) iff b = 0 by choice of V1.Thus, (I − M)V1b = 0 iff b = 0; hence (I − M)V1 has full column rank and V1(I − M)V1 is
Exercise 10.5 Prove observations (a), (b), and (c).
Exercise 10.4 Ferguson (1967, Section 2.7, page 74) proves the following:
Exercise 10.3 The BLUE of Xβ can be obtained by taking T = V + XX. Prove this by showing that(a) C(X) ⊂ C(T ) if and only if TT−X = X, and
Exercise 10.2 Show that if Λ = PX and if A0Y is a BLUE of Xβ, then PA0Y is a BLUE of Λβ.
Exercise 10.1 Show that A0Y is a BLUE of Xβ if and only if, for every estimable function λβ such that ρX = λ, ρA0Y is a BLUE of λβ.
Exercise 9.6.4 State whether each design given below is a balanced incomplete block design, and if so, give the values ofb, t, k, r , and λ.(a) The experiment involves 5 treatments: A, B, C, D, and E. The experiment is laid out as follows.Block Treatments Block Treatments 1 A, B,C 6 A, B, D 2 A,
Exercise 9.6.3 The missing value procedure that consists of analyzing the model (Y − Zγˆ ) = Xβ + e has been shown to give the correct SSE and BLUEs;however, sums of squares explained by the model are biased upwards. For a randomized complete block design with a treatments and b blocks and the
Exercise 9.6.2 Suppose that in Exercise 7.7.1 on motor oil pricing, the observation on store 7, brand H was lost. Treat the stores as blocks in a randomized complete block design. Plug in an estimate of the missing value and analyze the data without correcting the MSTrts or any variance estimates.
Exercise 9.6.1 Sulzberger (1953) andWilliams (1959) examined themaximum compressive strength parallel to the grain (y) of 10 hoop trees and howitwas affected by temperature.Acovariate, the moisture content of thewood (x),was alsomeasured.Analyze the data, which are reported below Temperature in
Exercise 8.6.2 Show that the set of indices i = 1, . . . ,a, j = 1, . . . ,a, and k =(i + j + a − 1) mod(a) determines a Latin square design.Hint: Recall that t mod(a) means t modulo a and is defined as the remainder when t is divided by a.
Exercise 8.6.1 A study was performed to examine the effect of two factors on increasing muscle mass in weight lifters. The first factor was dietary protein level.The levels were use of a relatively low protein diet (L) and use of a high protein diet(H). The second factor was the use of anabolic
Exercise 7.7.7 Assuming that the reader is familiar with the analysis of twoway tables of counts and treating the Ni j s as a two-way table of counts, show that proportional numbers hold if and only if the Ni j fit the model of independence perfectly which occurs if and only if all of the odds
Exercise 7.7.6 For Example 7.6.1, develop a test for H0 : A10 + A11 = A20 +A21.
Exercise 7.7.5 Consider the balanced two-way ANOVA with interaction model yi jk = μ + αi + ηj + γi j + ei jk, i = 1, . . . ,a, j = 1, . . . ,b, k = 1, . . . , N, with ei jks independent N(0, σ2). Find E[Y ( 1 n J n n+ Mα)Y ] in terms of μ, the αi s, theηj s, and the γi j s.
Exercise 7.7.4 Consider the linear model yi j = μ + αi + ηj + ei j , i = 1, . . . , a, j = 1, . . . ,b. As in Section 1, write X = [X0, X1, . . . , Xa, Xa+1, . . . , Xa+b]. If we write the observations in the usual order, we can use Kronecker products to write the model matrix. Write X = [J,
Exercise 7.7.3 Consider the model yi jk = μ + αi + ηj + γi j + ei jk, i = 1, 2, 3, 4, j = 1, 2, 3, k = 1, . . . , Ni j ,where for i = 1 = j , Ni j = N, and N11 =2N. This model could arise from an experimental design having α treatments of No Treatment (NT), a1, a2, a3 and η treatments of
Exercise 7.7.2 An experiment was conducted to examine thrust forces when drilling under different conditions. Data were collected for four drilling speeds and three feeds.Speed Feed 100 250 400 550 121 98 83 58 124 108 81 59 0.005 104 87 88 60 124 94 90 66 110 91 86 56 329 291 281 265 331 265 278
Exercise 7.7.1 In the mid-1970s, a study on the prices of various motor oils was conducted in (what passes for) a large town in Montana. The study consisted of pricing 4 brands of oil at each of 9 stores. The data follow.Brand Store P H V Q 1 87 95 95 82 2 96 104 106 97 3 75 87 81 70 4 81 94 91 77
Exercise 6.9.10 Show that in Section 2, γ∗ = β∗ and β0 = γ0 − (1/n)J n 1 Zγ∗.Hint: Examine the corresponding argument given in Section 1 for simple linear regression.
Exercise 6.9.9 Show that(a) ρ12·3 = ρ12 − ρ13ρ23 1 − ρ2 131 − ρ2 23(b) ρ12·34 = ρ12·4 − ρ13·4ρ23·4 1 − ρ2 13·41 − ρ2 23·4.
Exercise 6.9.8 Test whether the data of Example 6.2.1 indicate that the partial correlation coefficient ρy1·2 is different from zero.
Exercise 6.9.7 Test whether the data of Example 6.2.1 indicate that the multiple correlation coefficient is different from zero.
Exercise 6.9.6 Find E[y − ˆE (y|x)]2 in terms of the variances and covariances of x and y. Give a “natural” estimate of Ey − ˆE (y|x)2.
Exercise 6.9.5 Consider the model yi = β0 + β1xi + ei , ei s i.i.d. N(0, σ2), i = 1, 2, 3, . . . , n. If the xi s are restricted to be in the closed interval [−10, 15], determine how to choose the xi s to minimize(a) Var( ˆ β0).(b) Var( ˆ β1).(c) How would the choice of the xi s change if
Exercise 6.9.4 Consider the model yi = β0 + β1xi + ei , ei s i.i.d. N(0, σ2/wi ), i = 1, 2, 3, . . . , n, where the wi s are known numbers. Derive algebraic formulas forˆ β0, ˆ β1, Var( ˆ β0), and Var( ˆ β1).
Exercise 6.9.3 Two-Phase Linear Regression.Consider the problem of sterilizing canned pudding. As the pudding is sterilized by a heat treatment, it is simultaneously cooked. If you have ever cooked pudding, you know that it starts out soupy and eventually thickens. That, dear reader, is the point
Exercise 6.9.2 Maximizing a Quadratic Response.Consider themodel, yi = β0+β1xi+β2x2 i+ei , ei s i.i.d. N(0, σ2), i = 1, 2, 3, . . . , n.Let x0 be the value at which the function E(y) = β0 +β1x +β2x2 is maximized (or minimized).(a) Find the maximum likelihood estimate of x0.(b) Find a
Exercise 6.9.1 Calibration.Consider the regression model Y = Xβ +e, e ∼ N(0, σ2 I ) and suppose that we are interested in a future observation, say y0, that will be independent of Y and have mean x0β. In previous work with this situation, y0 was not yet observed but the corresponding vector
Exercise 6.9.10 Show that in Section 2, γ∗ = β∗ and β0 = γ0 − (1/n)J n 1 Zγ∗.Hint: Examine the corresponding argument given in Section 1 for simple linear regression.
Exercise 6.9.9 Show that(a) ρ12·3 = ρ12 − ρ13ρ23 1 − ρ2 131 − ρ2 23(b) ρ12·34 = ρ12·4 − ρ13·4ρ23·4 1 − ρ2 13·41 − ρ2 23·4.
Exercise 6.9.8 Test whether the data of Example 6.2.1 indicate that the partial correlation coefficient ρy1·2 is different from zero.
Exercise 6.9.7 Test whether the data of Example 6.2.1 indicate that the multiple correlation coefficient is different from zero.
Exercise 6.9.6 Find E[y − ˆE (y|x)]2 in terms of the variances and covariances of x and y. Give a “natural” estimate of Ey − ˆE (y|x)2.
Exercise 6.9.5 Consider the model yi = β0 + β1xi + ei , ei s i.i.d. N(0, σ2), i = 1, 2, 3, . . . , n. If the xi s are restricted to be in the closed interval [−10, 15], determine how to choose the xi s to minimize(a) Var( ˆ β0).(b) Var( ˆ β1).(c) How would the choice of the xi s change if
Exercise 6.9.4 Consider the model yi = β0 + β1xi + ei , ei s i.i.d. N(0, σ2/wi ), i = 1, 2, 3, . . . , n, where the wi s are known numbers. Derive algebraic formulas forˆ β0, ˆ β1, Var( ˆ β0), and Var( ˆ β1).
Exercise 6.9.3 Two-Phase Linear Regression.Consider the problem of sterilizing canned pudding. As the pudding is sterilized by a heat treatment, it is simultaneously cooked. If you have ever cooked pudding, you know that it starts out soupy and eventually thickens. That, dear reader, is the point
Exercise 6.9.2 Maximizing a Quadratic Response.Consider themodel, yi = β0+β1xi+β2x2 i+ei , ei s i.i.d. N(0, σ2), i = 1, 2, 3, . . . , n.Let x0 be the value at which the function E(y) = β0 +β1x +β2x2 is maximized (or minimized).(a) Find the maximum likelihood estimate of x0.(b) Find a
Exercise 6.9.1 Calibration.Consider the regression model Y = Xβ +e, e ∼ N(0, σ2 I ) and suppose that we are interested in a future observation, say y0, that will be independent of Y and have mean x0β. In previous work with this situation, y0 was not yet observed but the corresponding vector
Exercise 5.8.4 Suppose that in a balanced one-way ANOVA the group means¯y1·, . . . , ¯yt· are not independent but have some nondiagonal covariance matrix V.How can Tukey’s HSD method be modified to accommodate this situation?Exercise 5.8.5 For an unbalanced one-wayANOVA, give the contrast
Exercise 5.8.3 Compare all pairs of means in the coat–shirt exercise of Chapter 4.Use all of the appropriate multiple comparison methods discussed in this chapter to control the experimentwise error rate at α = 0.05 (or thereabouts).
Exercise 5.8.2 Test whether the four orthogonal contrasts you chose for the blue jeans exercise of Chapter 4 equal zero. Use all of the appropriate multiple comparison methods discussed in this chapter to control the experimentwise error rate at α = 0.05 (or thereabouts).
Exercise 5.8.1 Compare all pairs of means for the blue jeans exercise ofChapter 4.Use the following methods:(a) Scheffé’s method, α = 0.01,(b) LSD method, α = 0.01,(c) Bonferroni method, α = 0.012,(d) Tukey’s HSD method, α = 0.01,(e) Newman–Keuls method, α = 0.01.
Exercise 3.9.8 Under the conditions of Exercise 3.9.7, showthat (Y − Xb1)(M −M0)(Y − Xb1) = (Y − Xb2)(M − M0)(Y − Xb2).
Exercise 3.9.7 Consider a linear model Y = Xβ +e, E(e) = 0 with the (not necessarily estimable) linear constraint Λβ =d. Consider two solutions to the constraint, b1 and b2, so that Λbk =d, k = 1, 2. Define appropriate least squares fitted values Yˆk from the model Y = X0γ + Xbk + e where
Exercise 3.9.6 Discuss how the testing procedures from this chapter would change if you actually knew the variance σ2.
Exercise 3.9.5 Consider the model Y = Xβ +e, E(e) = 0, Cov(e) = σ2 I, (1)with the additional restrictionΛβ = d, where d = Λb for some (known) vector b and Λ = PX. Model (1) with the additional restriction is equivalent to the model(Y − Xb) = (M − MMP)γ +e. (2)If the parameterization
Exercise 3.9.4(a) Using the notation of Section 3.3 and in particular of model (3.3.6), show that the least squares estimate of β subject to the constraintΛβ = d is βˆ = Uγˆ + b.(b) What happens to the test of Λβ = d if Λβ has no estimable part?
Exercise 3.9.3 Consider a set of seemingly unrelated regression equations Yi = Xiβi + ei , ei ∼ N0, σ2 I, i = 1, . . . , r , where Xi is an ni × p matrix and the ei s are independent. Find the test for H0 : β1 = · · · = βr .
Exercise 3.9.2 For a standard linear model, find the form of the generalized likelihood ratio test of H0 : σ2 = σ2 0 versus HA : σ2 = σ2 0 in terms of rejecting H0 when some function of SSE/σ 2 0 is small. Show that the test makes sense in that it rejects for both large and small values of
Exercise 3.9.1 Consider themodel yi = β0 + β1xi1 + β2xi2 + ei , ei s i.i.d. N(0, σ2).Use the data given below to answer (a) and (b).Obs. 1 23 4 5 6 y −2 72 5 8−1 x1 4 −1 2 0 −2 3 x2 2 −3 0 −2 −4 1(a) Find SSR(X1, X2|J ) = R(β1, β2|β0).(b) Are β0, β1, and β2 estimable?
Exercise 2.11.8 Use Theorem 2.2.4 and its proof to show that there is a unique least squares estimate in C(X), say ˆ β0, and that all other least squares estimates are obtained by ˆ β = ˆ β0 + v for some v ⊥ C(X).
Exercise 2.11.7 For V positive definite andC(VX) ⊂ C(X) use PropositionB.12 and the results in Section B.4 to show that r (VX) = r (VXX) = r (XX) = r (X).Hint: Show N (VXX) = N (XX).
Exercise 2.11.6 Consider the linear modelY1 Y2=JN1 JN2μ +e, E(e) = 0, Cove1 e2=σ2 1 IN1 0 0 σ2 2 IN2.Find the BLUE ˆμand Var( ˆ μ) assuming that the variance parameters are known.How does the result change if σ2 1 is unknown but σ2 2 /σ 2 1 is known?
Exercise 2.11.5 Consider the model yi = β0 + β1xi1 + β2xi2 + β11x2 i1+ β22x2 i2+ β12xi1xi2 + ei , where the predictor variables take on the following values.i 1 2 3 4 5 6 7 xi1 1 1 −1 −1 0 0 0 xi2 1 −1 1 −1 0 0 0 Show that β0, β1, β2, β11 + β22, β12 are estimable and find
Exercise 2.11.4 Consider themodel yi = β1xi1 + β2xi2 + ei , ei s i.i.d. N(0, σ2).There are 15 observations and the sum of the squared observations is Y Y = 3.03.Use the normal equations given below to answer parts (a) through (c).(a) Estimate β1, β2, and σ2.(b) Give 98% confidence intervals
Exercise 2.11.3 Consider themodel yi = β1xi1 + β2xi2 + ei , ei s i.i.d. N(0, σ2).Use the data given below to answer (a) through (d). Show your work, i.e., do not use a regression or general linear models computer program.(a) Estimate β1, β2, and σ2.(b) Give 95% confidence intervals for β1
Exercise 2.11.2 Consider the model Y = Xβ + Xb +e, E(e) = 0, Cov(e) = σ2 I, where b is a known vector and Xb is often called an offset. Show that Proposition 2.1.3 is not valid for this model by producing a linear unbiased estimate of ρXβ, say a0 + aY , for which a0 = 0.Hint: Modify ρMY.
Exercise 2.11.1 Prediction. Consider a regression model Y=Xβ +e, e ∼ N(0,σ2 I ) and suppose that we want to predict the value of a future observation, say y0, that will be independent of Y and be distributed N(x0β, σ2).(a) Find the distribution of y0 − x0ˆ βMSE1 + x0(XX)−1x0
Exercise 1.5.8 Show that(a) for Example 1.0.1 the perpendicular projection operator onto C(X) is M = 1 6 J 6 6 + 1 70 ⎡⎢⎢⎢⎢⎢⎢⎣25 15 5 −5 −15 −25 15 9 3 −3 − 9 −15 5 3 1 −1 − 3 − 5 − 5 −3 −1 1 3 5 −15 −9 −3 3 9 15 −25 −15 −5 5 15 25(b) for Example
Exercise 1.5.7 Let A be symmetric, Y ∼ N(0, V), and w1, . . . ,ws be independentχ2(1) random variables. Show that for some value of s and some numbers λi , Y AY ∼s i=1 λiwi . Hint: Y ∼ QZ so Y AY ∼ ZQAQZ.Write QAQ = PD(λi )P.
Exercise 1.5.6 Let Y = (y1, y2, y3) have a N(μ, σ2 I ) distribution. Consider the quadratic forms defined by the matrices M1, M2, and M3 given below.(a) Find the distribution of each Y MiY .(b) Show that the quadratic forms are pairwise independent.(c) Show that the quadratic forms are mutually
Exercise 1.5.5 Let Y = (y1, y2) have a N(0, I ) distribution. Show that if A =1 a a 1B =1 b b 1, then the conditions of Theorem 1.3.7 implying independence of Y AY and Y BY are satisfied only if |a| = 1/|b| and a = −b.What are the possible choices for a and b?
Exercise 1.5.4 Let Y ∼ N(Jμ, σ2 I ) and let O =n−1/2 J, O1be an orthonormal matrix.(a) Find the distribution of OY .(b) Show that ¯y· = (1/n)J Y and that s2 = Y O1O1Y/(n − 1).(c) Show that ¯y· and s2 are independent.Hint: Show that Y Y = Y OOY = Y (1/n)J JY + Y O1O1Y .
Exercise 1.5.3 The density of Y = (y1, y2, y3) is(2π)−3/2|V|−1/2e−Q/2, where Q = 2y2 1+ y2 2+ y2 3+ 2y1 y2 − 8y1 − 4y2 + 8.Find V−1 and μ.
Exercise 1.5.2 Let Y = (y1, y2, y3) with Y ∼ N(μ, V ), whereμ = (5, 6, 7)and V =⎡⎣2 0 1 0 3 2 1 2 4⎤⎦.Find(a) the marginal distribution of y1,(b) the joint distribution of y1 and y2,(c) the conditional distribution of y3 given y1 = u1 and y2 = u2,(d) the conditional distribution of y3
Exercise 1.5.1 Let Y = (y1, y2, y3) be a random vector. Suppose that E(Y ) ∈M, whereM is defined by M = {(a, a −b, 2b)|a, b ∈ R}.(a) Show thatM is a vector space.(b) Find a basis forM.(c) Write a linear model for this problem (i.e., find X such that Y = Xβ + e, E(e) = 0).(d) If β = (β1,
Exercise 4.1 To develop some facility with this notation, let Tr = [ti j], where ti j =δir − Nr nfor r = 1, . . . ,t. Find Tr Tr, Tr Ts for s = r, and JTr.A very important application of this notation is in characterizing the vector MY.As discussed in Section 3.1, the vector MY is the base
Exercise 4.2 Show that MY = [ti j], where ti j = ¯ yi·and MαY = [ui j], where ui j = ¯ yi·− ¯ y·· .Hint: Write MαY = MY −( 1 n Jn n )Y.Knowing these characterizations MY and MαY tell us how to find Mv and Mα v for any vector v. In fact, they completely characterize the perpendicular
Exercise 4.3 Verify that the estimate of μ +αi is ¯ yi· and that the algebraic formulas for the sums of squares in the ANOVA table are correct.Hint: To find, for example, Y (M−[1/n]Jn n )Y =YMαY, use Exercise 4.2 to get MαY and recall that YMαY = [MαY] [MαY].
Exercise 4.4 Verify that the formulas for expected mean squares in the ANOVA table are correct.
Exercise 4.5 Show that α1 =α2 = · · · =αt if and only if all contrasts are zero.
Exercise 4.6 Using the theories of Sections 3.3 and 2.6, respectively, find the F test and the t test for the hypothesis H0 : Σt i=1λiαi = d in terms of the MSE, the ¯ yi·s, and the λis.
Exercise 4.7 Suppose N1 = N2 = · · · = Nt ≡ N. Rewrite the ANOVA table incorporating any simplifications due to this assumption.
Exercise 4.8 If N1 = N2 = · · · = Nt ≡ N, show that two contrasts λ 1β and λ 2βare orthogonal if and only if λ 1λ2 = 0.
Exercise 4.9 Find the least squares estimates of μ, α1, and αt using the side condition α1 = 0.
Exercise 4.10 Using ρ as defined by (1) and X as defined in Section 1, especially(4.1.1), show that(a) ρX =λ , where λ = (0,λ1, . . . ,λt ).(b) ρ ∈ C(X).

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