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4. (10%) Consider the matrixwith parameter x. Specify all numbers x, if any, for which A is positive definite. Explain your answer.

其他申論題

(b) (5%) If A is row equivalent to the identity matrix and AB = AC, then B must equal C.

(a) (5%) Use the Gram-Schmidt process to find an orthonormal basis for the column space of A.

(b) (7%) Factor A into a product QR, where Q has an orthonormal set of column vectors and R is upper triangular.

(c) (8%) Solve the least squares problem Ax = b

(a)(4%) Find the probability that the smallest number drawn is more than or equal to 4.

(b)(4%) Find the probability that 8 is the largest number drawn.

6.(8%) We have three coins; the first two coins are fair and the last coin is two-headed. We pick one of the coins at random, and toss it twice. Heads show both times. Find the probability that the coin picked is fair.

7.(8%) The random variable x is uniform in the interval [-5, 5]. Find C.D.F. Fy(y) and p.d.f. fy(y).

8.(10%) The joint p.df. of x and y for some k. Deter- mine the conditional p.d.f.

(a)(8%) Determine the joint p.d.f. of z and w.

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