| 4月28日 第2節 電子系丙組 | 誠實是我們珍視的美德， 我們喜愛「拒絕作弊，堅守正直」的你！ |
| 科目：工程數學 | （共2頁 第1頁） |
| 1. | Let P be any solution of AX=B, and Q is a solution of AX=0.
(a) What is the form of every solution of AX=B. (3%) (b) Please prove it. (7%) |
| 2. | Let and
AX=B. (a) Find the inverse of A (that is A-1). (7%) (b) Solve for the solution of X. (3%) |
| 3. | Consider the system:
 (a) Find the eigenvalues and eigenvectors of A. (10%) (b) Solve the solutions for x1, x2, and x3. (10%) |
| 4. | If is
the coordinate vector with respect to [e1, e2].
(a) Please find the transition matrix from [a1, a2] to [e1, e2]. (5%) (b) Please find the coordinates of x with respect to [a1, a2]. (5%) |
| 5. | Suppose the number of earthquakes that occur in T
weeks in Taiwan is a Poisson random variable, N, with expected
value λT.
Find the probability that at least 3 earthquakes occur during the next
2 weeks. (10%) |
| 6. | Let X and Y be positive independent random variables with
the identical probability density function ae-x for
x > 0 and ae-y for y > 0, respectively, where a
is a constant. Find the constant a and the joint probability density
function of U= X/Y and V= X+Y. (20%) |
| 7. | Show the following two fundamental inequalities in probability
theory. (a) If X is a random variable that takes only nonnegative values, then for any value b > 0,  
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