113 工程數學(D)¶
說明¶
- \(\mathbb{R}\)：實數集
- \(\mathbb{C}\)：複數集
- \(T\)：轉置
- b：行向量
- Column space：\(A\) 的所有線性組合
- Nullspace：\(A\mathbf{x} = \mathbf{0}\) 的所有解
- Row space = \(A^T\) 的 column space
- Left nullspace = \(A^T\) 的 nullspace
第 1 題（25%）— 機率¶
At noon on a particular day, an experiment is conducted to record new call attempts at a telephone switchboard. Random variable \(X\) denotes the arrival time of the first call, as measured by the number of seconds after noon, and random variable \(Y\) denotes the arrival time of the second call. It is found that \(X\) and \(Y\) are continuous random variables with joint PDF as follows:
where \(\lambda > 0\) calls/second is a constant.
(a) (6%) Derive the marginal PDFs of the arrival times of the first and second calls respectively.
(b) (4%) Derive the expected values of the arrival times of the first and second calls respectively.
© (5%) Let \(W\) be the random variable denoting the time difference between the arrivals of the second call and the first call. Derive the PDF of \(W\).
(d) (5%) It is observed that the second call arrives at time \(y > 0\), but the arrival time of the first call is unknown. Derive the PDF of the arrival time of the first call under this observation.
(e) (5%) It is observed that the first call arrives at time \(x > 0\). At time \(x + \tau\), \(\tau > 0\), it is also observed that the second call still does not arrive yet. Derive the PDF of the arrival time of the second call under this observation.
我的答案¶
(a)
(b)
©
(d)
(e)
第 2 題（7%）— 機率¶
\(X_1\) and \(X_2\) are identically distributed random variables with expected value \(\mu\) and variance \(\sigma^2\). Let \(Y = X_1 + kX_2\) be another random variable, where \(k\) is a constant. Find the range of \(k\) such that the variance of \(Y\) is greater than or equal to \(\sigma^2\).
我的答案¶
第 3 題（18%）— 機率¶
Alice participates in a tournament, in which a series of games are played indefinitely until she loses. The scoring rule of the game is that the winner is granted 1 point, the loser is deducted 1 point, and a tie grants both players 0 points. Each game is played independently, and it is known that in each game the probability for Alice to win is 0.5 and to lose is 0.4. Let \(X\) be the total number of points Alice earns in the tournament.
(a) (8%) Derive the MGF \(\phi_X(s)\) of \(X\).
(b) (5%) Derive the variance of \(X\).
© (5%) Derive an upper bound of the probability that \(X\) is greater than or equal to 9 times of its expected value using the Chebyshev Inequality.
我的答案¶
(a)
(b)
©
第 4 題（5%）¶
Answer the following questions with "True" or "False".
(a) The vectors \([1, 3, 2]\) and \([2, 1, 3]\) and \([3, 2, 1]\) are dependent.
(b) The vectors \([1, -3, 2]\) and \([2, 1, -3]\) and \([-3, 2, 1]\) are dependent.
© If the row space equals the column space, then \(A^T = A\).
(d) If \(A^T = -A\), then the row space of \(A\) equals the column space.
(e) \(A\) and \(A^T\) have the same number of pivots.
(Getting 5 points if all answers are correct. Otherwise, 0 point.)
我的答案¶
(a)
(b)
©
(d)
(e)
第 5 題（5%）¶
Suppose columns of a matrix \(A\) are \(n\) vectors in \(\mathbb{R}^m\). Answer the following questions with "True" or "False".
(a) Let \(A\) be a real matrix. Then, \(A^T A\) is invertible if and only if the columns of \(A\) are linearly independent.
(b) The equation \(A\mathbf{x} = \mathbf{b}\) has a solution if and only if \(A\) has a pivot position in every row.
© Let \(n = 3\) and \(m = 5\). Then \(A\mathbf{x} = \mathbf{b}\) is not necessarily solvable.
(d) There exists a \(3 \times 3\) diagonal matrix \(A\) such that \(A \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix} = \begin{bmatrix} 4 \\ 5 \\ 6 \end{bmatrix}\).
(e) If \(A \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} x \\ 0 \end{bmatrix}\), then the null space of \(A\) is span \(\left\{ \begin{bmatrix} 1 \\ 0 \end{bmatrix} \right\}\).
(Getting 5 points if all answers are correct. Otherwise, 0 point.)
我的答案¶
(a)
(b)
©
(d)
(e)
第 6 題（5%）¶
Answer the following questions.
(a) \(A^T \mathbf{y} = \mathbf{b}\) is solvable when \(\mathbf{b}\) is in which of the four subspaces of \(A\)? ______
(b) As above, the solution \(\mathbf{y}\) is unique when the ______ contains only the zero vector.
© The dimension of the subspace of \(n \times n\) symmetric matrices is ______.
(d) Suppose the kernel of an \(5 \times 3\) matrix \(A\) is one dimensional. What is the dimension of \(A\)? ______
(e) (True or False) Suppose \(A\), \(B\) and \(C\) are \(2 \times 2\) matrices such that \(A\) commutes with \(B\) and \(B\) commutes with \(C\). Then, \(A\) commutes with \(C\).
(Getting 5 points if all answers are correct. Otherwise, 0 point.)
我的答案¶
(a)
(b)
©
(d)
(e)
第 7 題（5%）¶
Suppose \(A\) is the sum of two matrices of rank one: \(A = \mathbf{u}\mathbf{v}^T + \mathbf{w}\mathbf{z}^T\). Answer the following questions.
(a) Which vectors span the column space of \(A\)? ______
(b) Which vectors span the row space of \(A\)? ______
© The rank is less than 2 if ______ or if ______.
(d) What is the rank of \(A\) if \(\mathbf{u} = \mathbf{z} = [1, 0, 0]^T\) and \(\mathbf{v} = \mathbf{w} = [0, 0, 1]^T\)? ______
(Getting 5 points if all answers are correct. Otherwise, 0 point.)
我的答案¶
(a)
(b)
©
(d)
第 8 題（5%）¶
The matrix
has eigenvalues \(-48\) and \(24\). What is the third eigenvalue?
我的答案¶
第 9 題（10%）¶
Let
Calculate the power \(A^{2024}\).
我的答案¶
第 10 題（15%）¶
Find an orthonormal basis of polynomials on real lines with degree at most 2, where the inner product is given by \(\langle p, q \rangle = \int_{-1}^{1} p(x)q(x) \, dx\).