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### HW3

Course: STA 244, Spring 2008
School: Duke
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Word Count: 351

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3 Due STA244 2/5/2005 Homework 2/12/2001 1. Recall from class that a non-central 2 (m, ) can be represented as a Poisson mixture of central 2 random variables, where Y P (/2) and X|Y 2 (m + 2y, 0). Find the mean and variance of a non-central Chi-squared random variable with m degrees of freedom and non-centrality parameter . 2. Consider the linear model Y = + N (0, 2 In ) where = X, X is n p rank r p...

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3 Due STA244 2/5/2005 Homework 2/12/2001 1. Recall from class that a non-central 2 (m, ) can be represented as a Poisson mixture of central 2 random variables, where Y P (/2) and X|Y 2 (m + 2y, 0). Find the mean and variance of a non-central Chi-squared random variable with m degrees of freedom and non-centrality parameter . 2. Consider the linear model Y = + N (0, 2 In ) where = X, X is n p rank r p and (a) Show that if X is of full column rank r = p, then PX = X(X X)-1 X is an orthogonal projection onto the space spanned by the columns of X. (b) Find an expression for the projection PX in the non-full rank case (hint: use the Singular Value Decomposition Thm with X). (c) Show that QX I - PX is also an orthogonal projection on to the orthogonal complement of the span of X, S(X) . (d) Find the distribution of ||PX Y ||2 . (general rank case) (e) Find the distribution of ||QX Y ||2 . (general rank case) (f) Find the distribution of Y ||PX ||2 /||QX Y ||2 ). 3. Suppose we have a n p matrix Q and a p p upper triangular matrix R such that Q Q = Ip and QR = X. assume that X is of rank p (a) Show that R R = X X. ^ ^ (b) Show that = R-1 Q Y . Thus to compute , one computes z = Q Y then solves ^ the system of equations R = z by back substitution without explicit inversion of X X. (c) Show that QQ is an orthogonal projection of rank p onto the S(X) and that ^ Y = QQ Y . (d) Find e (the residuals) and residual sum of squares in terms of Y and Q. ^ (e) Show that the variance of a linear combination ...

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Duke - STA - 103
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Duke - STA - 103
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Duke - STA - 103
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Duke - STA - 244
STA2444/9/2002Homework 7Due 4/16/2001 1. Problem 15.7 in CW. To obtain case diagnostics in S-Plus, fit a model using the QR option, i.e. mylm.obj &lt;- lm(Y X1 + X2, data=mydataframe, qr=T) To obtain the case diagnostics, use the function ls.diag(
Duke - STA - 103
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Duke - STA - 103
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Duke - STA - 103
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Duke - STA - 244
STA2443/19/2005Homework 6Due 3/28/2002 1. For the usual linear model Y N (X, -1 In ) with prior distributions N (bo , Vo ) independent of and p() 1/: (a) Find the posterior distribution of |. (b) Can you find a closed form expression for th
Duke - STA - 102
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Duke - STA - 102
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Duke - STA - 102
subject fev1 gender 1 2.30 0 2 2.15 1 3 3.50 1 4 2.60 0 5 2.75 0 6 2.82 1 7 4.05 1 8 2.25 1 9 2.68 0
Duke - STA - 244
x1 x2 y1 y2 y3 y4 10 8 8.04 9.14 7.46 6.58 8 8 6.95 8.14 6.77 5.76 13 8 7.58 8.74 12.74 7.71 9 8 8.81 8.77 7.11 8.84 11 8 8.33 9.26 7.81 8.47 14 8 9.96 8.1 8.84 7.04 6 8 7.24 6.13 6.08 5.25 4 19 4.26 3.1 5.39 12.5 12 8 10.84 9.13 8.15 5.56
Duke - STA - 244
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Duke - STA - 244
FUEL/POP INC LIC/POP POP TAX VEH/POP VM/VEH 644.147 14.826 0.70923 4041 13 0.911408 11.0684 474.545 21.761 0.549091 550 8 0.669091 10.5625 552.524 16.297 0.660573 3665 18 0.777899 12.2119 683.539 14.218 0.735857 2351 18.7 0.615908 14.0981 501.34
Duke - STA - 244
Pressure Temp 20.79 194.5 20.79 194.3 22.4 197.9 22.67 198.4 23.15 199.4 23.35 199.9 23.89 200.9 23.99 201.1 24.02 201.4 24.01 201.3 25.14 203.6 26.57 204.6 28.49 209.5 27.76 208.6 29.04 210.7 29.88 211.9 30.06 212.2
Duke - STA - 244
Duke - STA - 244
STA2444/18/2002Homework 8Due 4/26/2001 Refer to Exercise 11.5 in CW (page 285). Use any appropriate methods covered in class to answer the problem (Bayesian, Frequentist, or compare both). Provide a typed solution describing the problem and how
Duke - STA - 244
STA2442/14/2005Homework 4Due 2/21/2001 1. Consider the linear model Y = X 1 1 + X 2 2 + where X1 is n q and X2 is n (p q), with both matrices of full column rank. Consider the problem of testing N H : 1 = 0. Assume that N (0, 2 In ). (a) Gi
Duke - STA - 244
U X1 X2 Y 0.493151 1 1 0.872302 1.40245 2 1 1.59988 2.31175 3 1 2.4019 3.22104 4 1 3.25942 4.13034 5 1 4.14616 5.03964 6 1 5.04607 5.94894 7 1 5.95154 6.85823 8 1 6.85928 7.76753 9 1 7.76795 8.67683 10 1 8.677 0.0770038 1 2 0.557762 0.986
Duke - STA - 113
Name:Section:STAT 113 Midterm 31 Otis 1979, Journal of Psychology interviewed people waiting to see the space aliens lm Close Encounters of the Third Kind.&quot; Each person was asked to state his or her degree of agreement with the statement Life on
Duke - STA - 113
Name:Section:STAT 113 Midterm 21a. 1pt Suppose y is a normally distributed random variable with mean 0 and variance 1.0, i.e. y is standard normal. Find P ,1:0 y 0:5.1b. 2pt Suppose y is normally distributed random variable with mean 10 and va
Duke - STA - 113
Homework 9a SolutionsAs yi iid Bernoullip, then E yi = = p. Setting this expression to its respective sample P y =1 ^ y moment, we obtain: = n , or p = n ^ n! y n,y where K = 8.8 c. For the Binomial experiment, the likelihood function is L = K p
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Name:1a. 2pt Suppose y is normally distributed random variable with mean = 5:0 and variance 2 = 4:0, i.e. y N 5:0; 4:0. Find P 3:0 y 12:0. 1b. 2pt Suppose y is a 2 distributed random variable with = 12 degrees of freedom. Find cuto s c and d, su
Duke - STA - 113
Name:Section:STAT 113 Midterm 31 Otis1 1979 interviewed people waiting to see the space aliens lm Close Encounters of the Third Kind.&quot; Each person was asked to state his or her degree of agreement with the statement Life on Earth is being observ
Duke - STA - 113
Name:Section:STAT 113, Spring 99 Midterm 3On all problems, please show your work. Just the correct answer without justi cation and intermediate results is not acceptable.Note:1.In a survey of college students, it was found that X = 69 of th
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SOLUTIONNote: Version B had slightly di erent numbers. But the basic problems were the same. You can recognize Version B by Name&quot; instead of Name:&quot; on the top line, i.e., a missing :&quot; after Name&quot;. 1. On questions 1a-f: 2pts for the correct choice;
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