Academics

QUIZ 1 Probability and Random Variables 2026.09.28 **Problem 1. [20 Points]** A fair six-sided die has its faces labeled $\{1, 2, 2, 3, 3, 3\}$. The die is rolled eight times independently. i. (10 pts) Find the probability that the number 1 appears exactly twice and the number 2 appears exactly three times. ii. (10 pts) Given that the number 3 appears exactly four times, find the conditional probability that the number 1 appears exactly twice. --- **Solution for Problem 1** **Concept Overview**: This problem tests the **Multinomial Distribution** for independent repetitions of a discrete experiment with more than two possible outcomes, as well as **Conditional Probability**. First, determine the single-trial probabilities fo...

The inner product is a core algebraic structure shared across linear algebra, functional analysis, signal processing, machine learning, and quantum mechanics. Taking the inner product as its sole thread, this paper begins with the dot product in finite-dimensional Euclidean space and progressively introduces the axioms of inner product spaces, orthogonal decomposition, least-squares projection, Hilbert spaces, Fourier series and transforms, convolution, the discrete cosine transform, wavelet analysis, self-attention mechanisms, kernel methods, and state-vector projection in quantum mechanics. It reveals the mathematical unity underlying these seemingly disparate concepts: define an inner product → establish an orthonormal basis → project and decompose → extract information. This paper aims to provide readers with a cognitive map that cuts across mathematics, engineering, and physics.

Introduction This article systematically elaborates on the impedance characteristics and physical significance of resistors, capacitors, and inductors in alternating current (AC) circuits. Through phasor methods and frequency-domain analysis, the impedance formulas for these three components are derived, revealing the phase relationships between voltage and current as well as the patterns of energy variation. Furthermore, the mathematical expressions and frequency characteristics of resistance, capacitive reactance, and inductive reactance are compared and summarized. This facilitates an intuitive understanding of classic principles such as "passing AC while blocking DC" and "passing DC while blocking AC," providing a theoretical foundation for subsequent applications including filte...