(b) If the first moment and second moment of \(X\) are 2 and 20, respectively, find the moment-generating function \(M_Z(t)\) of the random variable \(Z = \frac{(X+6)}{2}\) based on the result obtained from (a).

(a) Show that the input \(n\) signals are linearly independent.

(b) Please verify that if this mixer performs a linear transformation in this system.

(a) If the period of a set of periodic functions is \(2\pi\), what is the orthonormal basis of the vector space that these periodic functions belong to? You have to show the vectors in the basis are orthonormal to get full credit.

(b) From the spectral theorem, find a vector \(g(x)\) in a subspace spanned by only \(\{1, \cos(x), \cos(2x), \sin(x), \sin(2x)\}\) so that \(g(x)\) has the shortest distance to the function \(f(x) = x\) in the interval \([-\pi, \pi]\).Note: In this problem, you may need: \(\int x \sin ax dx = -\frac{x}{a} \cos ax + \frac{1}{a^2} \sin ax\) and \(\int x \cos ax dx = \frac{x}{a} \sin ax + \frac{1}{a^2} \cos ax\)

(a) The set \(S \triangleq \{ \mathbf{A} \mid \mathbf{A} \in \mathbb{R}^{n \times n}, \mathbf{A}^T = -\mathbf{A} \}\) is not a subspace.

(b) Let \(\mathbf{A}, \mathbf{B} \in \mathbb{R}^{n \times n}\). Suppose the system \(\mathbf{A}\mathbf{x} = \mathbf{0}_n\) has infinitely many solutions and \(\mathbf{B}\mathbf{x} = \mathbf{0}_n\) has only one solution. This implies the system \(\mathbf{A}\mathbf{B}\mathbf{x} = \mathbf{0}_n\) has exactly one solution.

(c) If \(\mathbf{A}\) has an eigenvector \(\mathbf{x}\) with eigenvalue \(\lambda\), then \(\exp(\mathbf{A})\) has \(\mathbf{x}\) as an eigenvector with the eigenvalue \(\exp(\lambda)\).

(d) If \(\mathbf{A}, \mathbf{B} \in \mathbb{R}^{n \times n}\) and \(\mathbf{A}\mathbf{B} - \mathbf{B}\mathbf{A} = \mathbf{A}\), then \(\mathbf{A}\) is singular.

(e) Consider the following linear equations characterized by \(\mathbf{A} \in \mathbb{R}^{m \times n}\):\[\mathbf{A}\mathbf{x} = \mathbf{b}.\]Then \(\widehat{\mathbf{x}} = \mathbf{A}^{\dagger}\mathbf{b} + \mathbf{v}\) is a solution to the above linear equations, where \(\mathbf{A}^{\dagger}\) is the pseudo-inverse of \(\mathbf{A}\) and \(\mathbf{v} \in \mathcal{N}(\mathbf{A})\) (null space of \(\mathbf{A}\)). If your answer is "False", please describe the physical meaning of \(\mathbf{A}\widehat{\mathbf{x}}\).

(a) Consider a new binary random variable \(A\) which is the binary sum of \(a_1\) and \(a_2\), i.e., \(A = a_1 \oplus a_2\). Please show that the log-likelihood ratio of \(A\) defined as , can be expressed as .

110年 - 110 國立中央大學_碩士班招生考試_機械工程學系、機械工程學系(光機電工程)、能源工程研究所_(共同)：工程數學#125285

110年 - 110 國立臺灣大學_碩士班招生考試_部分系所：工程數學(A)#125255

110年 - 110 台灣聯合大學系統_碩士班招生考試_電機科:工程數學(C)#125211

110年 - 110 台灣聯合大學_碩士班招生_電機類：工程數學C#124898

110年 - 台聯大_工程數學B#113346

110年 - 110 國立臺灣科技大學_碩士班招生試題_材料科學與工程系(乙組)：工程數學#113288

110年 - 110 國立台北科技大學碩士班招生考試_自動化科技研究所：工程數學#111318

110年 - 110 國立臺灣大學_碩士班招生考試_部分系所:工程數學(D)(含線性代數、機率與統計)#111303

110年 - 110 台灣聯合大學系統碩士班招生考試_電機類:工程數學A#110408

110年 - 110 國立臺北科技大學_碩士班招生考試_機械工程系機電整合碩士班：工程數學#107430