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Tutor profile: Chao Z.

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Chao Z.
Very professional statistics Ph.d.
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Subject: Linear Algebra

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Question:

Let A= −1 2 0 1 a) Find the eigenvalues of A and their corresponding eigenvectors. b) Show that the eigenvectors forms a basis for R^2. c) Diagonalize matrix A if possible.

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Chao Z.
Answer:

Find the eigenvalues using the characteristic polynomial given by Det(A−λI) , where I is the identity matrix and Det is the determinant. Substitute A and I Det(A−λI)=Det([ −1 2 0 1 ]−λ[ 1 0 0 1 ]) =Det[ −1−λ 2 0 1−λ ] Evaluate the determinant =(−1−λ)(1−λ) Find the eigenvalues by solving the characteristic equation Det(A−λI)=0 Hence (−1−λ)(1−λ)=0 gives the eigenvalues: λ1=−1 , λ2=1 Find the eigenvalue corresponding to each eigenvector Eigenvector corresponding to λ=λ1=−1 The eigenvector x=[ x1 x2 ] corresponding to λ=−1 is the solution to the system (A−λ1I)[ x1 x2 ]=0 [ 0 2 0 2 ][ x1 x2 ]=0 Write the above system as an augmented matrix [ 0 2 | 0 0 2 | 0 ] Row reduce using Gauss-Jordan method [ 0 1 | 0 0 0 | 0 ] x1 is the free variable. x2=0 Vector x is given by x=x1[ 1 0 ] Eigenvector corresponding to λ=λ2=1 The eigenvector x=[ x1 x2 ] corresponding to λ=1 is the solution to the system (A−λ1I)[ x1 x2 ]=0 [ −2 2 0 0 ][ x1 x2 ]=0 Write the above system as an augmented matrix [ −2 2 | 0 0 0 | 0 ] Row reduce using Gauss-Jordan method [ 1 −1 | 0 0 0 | 0 ] x2 is the free variable. x1=x2 Vector x is given by x=x2[ 1 1 ] Conclusion: the eigenvalues and their corresponding eigenvectors are given by: λ=−1,1 in the corresponding order {[ 1 0 ],[ 1 1 ]} b) To prove that the eigenvectors forms a basis for R2, it is enough to show that the two vectors are linearly independent. Let us use the test for linearity. the augmented matrix made up of the eigenvectors and zeros for the last column on the right as follows: [ 1 1 | 0 0 1 | 0 ] (I) Row reduce using Gauss-Jordan the above augmented matrix. [ 1 0 | 0 0 1 | 0 ] The system corresponding to the augmented matrix (I) above has one solution only given by [ 0 0 ] and therefore the eigenvectors are linearly independent and hence the two eigenvectors form a basis for R2. c) Matrix A is diagonalizable and we need to find matrix P and its inverse P−1 We now construct a matrix P whose columns are the eigenvectors as follows P=[ 1 1 0 1 ] Find P−1 as follows: Write the augmented matrix A|I , where I is the identity matrix [ 1 1 | 1 0 0 1 | 0 1 ] Row reduce the above augmented matrix [ 1 0 | 1 −1 0 1 | 0 1 ] The above matrix has the form I|P−1 where P−1 is the inverse of matrix P and is given by P−1=[ 1 −1 0 1 ] Let D be the diagonal matrix whose entries in the main diagonal are the eigenvalues and P−1 be the inverse of matrix P. Matrix A may now be diagonalized as follows A=PDP−1=[ 1 1 0 1 ][ −1 0 0 1 ][ 1 −1 0 1 ] Use a calculator to check that the above diagonalization is correct. In fact it is enough to check that AP=PD which does not require the computation of P−1. Note that the order in which the eigenvectors are arranged in matrix P and the eigenvalues in matrix D is important. The eigenvector in column k in matrix P corresponds to the eigenvalue in row k in matrix D. Red and blue colors are used in the above example.

Subject: Calculus

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Question:

If f(x) = x 3 - 3x 2 + x and g is the inverse of f, then g '(3) is equal to (A) 10 (B) 1 / 10 (C) 1 (D) None of the above

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Chao Z.
Answer:

(B). Use g '(x) = 1 / f ' (g(x)) given as the answer to question 3 above to write g '(3) = 1 / f ' (g(3)). First, find g(3) which is the solution to the equation f(x) = 3 by definition of the inverse function. x 3 - 3x 2 + x = 3 The above equation has one real solution x = 3. g(3) = 3, the solution of the above equation. Then compute f '(x) = 3 x 2 - 6 x + 1. f ' (g(3)) = 3 (3) 2 -6 (3) + 1 = 10; and then substitute in the formula that gives g '(3) = 1 / 10.

Subject: Statistics

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Question:

Homer performs three simulation studies. His population is skewed to the right. For one study he has his computer generate 10,000 random samples of size n = 10 from the population. For each random sample, the computer calculates the Gosset 95% confidence interval for µ and checks to see whether the interval is correct. His second study is like his first, but n = 100. Finally, his third study is like the first, but n = 200. In one of his studies, Homer obtains 9,504 correct intervals; in another, he obtains 9,478 correct intervals, and in the remaining study he obtains 8,688 correct intervals. Based on what we learned in class, match each sample size to its number of correct intervals. Explain your answer.

Inactive
Chao Z.
Answer:

We know that sometimes Gosset does not work well, especially for a skewed population. But if Gosset performs poorly for a particular n, it will do better if n is increased. In this example, one performance is poor—the one with 8,688 correct CIs. This must be for n = 10. If Gosset performs well for a specific n, it will also perform well for any larger n. The 9,478 and 9,504 correct are both good performances; thus, we cannot tell which goes with n = 100 and which goes with n = 200

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