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110年 - 110 國立中央大學_碩士班招生考試_通訊工程學系/不分組(一般生)：工程數學#105325

科目：研究所、轉學考(插大)◆工程數學 | 年份：110年 | 選擇題數：0 | 申論題數：15

試卷資訊

所屬科目：研究所、轉學考(插大)◆工程數學

選擇題 (0)

申論題 (15)

1. (10%) Let A and B be 3 ✖3 matrices with det(A) = 5 and det(B) = -6. Find the value of: (a) (5%) det(2AB)

(b)(5%) 61cbc28123cd8.jpg
(a) (5%) If Ax = Bx for some nonzero vector x, then the matrices A and B must be equal.
(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

4. (10%) Consider the matrix61cbc3afb0d17.jpgwith parameter x. Specify all numbers  x, if any, for which A is positive definite. Explain your answer. 
(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]. 61cbc5292e152.jpgFind C.D.F. Fy(y) and p.d.f. fy(y).

8.(10%) The joint p.df. of x and y61cbc62c2b863.jpg

for some k. Deter-

mine the conditional p.d.f.61cbc5e5a63a9.jpg
(a)(8%) Determine the joint p.d.f. of z and w.
(b)(8%) Find the p.d.f. fz(z) and fw(w). Are z and w independent random variables?