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- Consider two random variables X and Y having joint density function f(x,y)=2^e−x−y,0
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- The Singular Value Decomposition (SVD) of a matrix R is given by USV^T. Consider an orthogonal matrix Q and A = QR. The SVD of A is given by U₁S₁V₁^T. Which of the following is/are true?
- Which of the following are true for any m × n matrix A of real numbers?
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- Let u be a n × 1 vector, such that u^T u = 1. Let I be the n × n identity matrix. The n × n matrix A is given by (I − kuu^T), where k is a real constant. u itself is an eigenvector of A, with eigenvalue -1. What is the value of k?
- Let A^n×n be a row stochastic matrix – in other words, all elements are non-negative and the sum of elements in every row is 1. Let b be an eigenvalue of A. Which of the following is true?
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- Let A^m×n be a matrix of real numbers. The matrix AA^T has an eigenvector x with eigenvalue b. Then the eigenvector y of A^T A which has eigenvalue b is equal to
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- Is the following a distribution function? F(x) = {0^e-2 e-1/x x>0 otherwise . If so, give the corresponding density function. If not, mention why it is not a distribution function.
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