Exercise [1.1.13] (a) Let G = ⋂ α Fα, with each Fα a σ-algebra. Since Fα a σ-algebra, we have that Ω ∈ Fα, and as this applies for all α, it follows that Ω ∈ G. Suppose now that A ∈ G. That is, A ∈ Fα for all α. Since each Fα is a σ-algebra, it follows that A ∈ Fα for all α, and hence A ∈ G. Similarly, let A = ⋃ iAi for some countable collection A1, A2, . . . of elements of G. By definition of ...

The trade-offs discussed in the lecture are clear: increasing the inversion level improves speed and area requirement at the expense of increased power consumption. Working in weak inversion helps lower the power consumption but requires larger area and has poor performance in terms of speed. The moderate inversion region exhibits a compromise between speed and area consumption with very little...

Notation. We will use some notation [mostly] from the class. A is the set of arms, T is the time horizon, K is the number of arms. At each round t, at is the arm chosen by the algorithm, and ct(a) is the cost of arm a. Total cost of arm a is cost(a) = ∑T t=1 ct(a). The best total cost for a subset S of arms is defined as cost∗(S) = infa∈S cost(a). The total cost of an algorithm ALG is cost(ALG)...

Clustering is an important machine learning task that tackles the problem of classifying data into distinct groups based on their features. An ideal clustering algorithm maximizes feature similarities within a cluster while minimizing the feature similarities across clusters. Some of the most common clustering algorithms include spectral clustering and k-means clustering. This project essential...

the purpose of using it in this homework. Namely, that the Fibonacci numbers, fn, count the tilings on a 2× (n− 1) board. First, suppose that n = 1. Then we have a 2× 0 board, and there is exactly 1 way to tile it. Suppose we have n = 2. Then, we have a 2× 1 board, and again there is exactly one way to tile it: with a single vertical domino. This provides the basis, where f1 = f2 = 1. Now, we n...

In distance learning missing direct communication between teachers and learners makes it difficult to provide direct assistance to students while they are solving their homework tasks. We address this problem especially for programming tasks and describe a system automatically analyzing students’ homework tasks, and providing understandable feedback. Our approach is adapted to the special situa...

Intelligent Tutoring Systems (ITS) have been proven to be efficient in providing students assistance and assessing their performance when they do their homework. Many research projects have been done to analyze how students’ knowledge grows and to predict their performance from within intelligent tutoring system. Most of them focus on using correctness of the previous question or the number of ...