Here are four questions:

1. Prove a standard Brownian motion is Gaussian process .

2. Prove a Brownian bridge is Gaussian process.

3. Prove Ornstein-Uhlenbeck process is Gaussian

4. Prove the position process is Gaussian.

Please provide as many detail as you can, thanks.

(Topology) Prove that the interior of a subset A of Xτ is the union of all τ -open sets contained in A.

Explanation:
Assume the reader understands derivatives, and knows the definition of instantaneous velocity (dx/dt), and knows how to calculate integrals but is struggling to understand them. Use students’ prior knowledge to provide an explanation that includes the concept and physical meaning of the integral of velocity with respect to time.

Reminder: The user is comfortable with the calculations, but is struggling with the concept. To fully address the prompt, emphasize the written explanation in English over the calculation

Prove that if a sequence is bounded, then it must have a convergent subsequence.

Given a set S = {1, -1, i, -i} where i² = -1 and with multiplication * on this set,

1. complete the following Cayley table:

(b) Determine whether, or not, the operation * is a binary operation on S.

(c) Is * commutative on S?

(d) Investigate the following properties of binary operations for this operation on S:

i. Closed

ii. Identity

iii. Inverse

iv. Associativity

Respond to the following prompts in a minimum of 175 words:

• Consider a recent interaction you have had with a co-worker or item that you saw in the media that might have included logical errors.
• Share the example as well as the logical errors that are present in the example. Be sure to use terms and concepts from the content covered this week to support your discussion.

Alice and Bob are have several piles of chips. On each turn they can either remove 1 or 2 chips from one pile, or split a pile into two nonempty piles. Players take turns and a player that cannot make a move loses. Find the value of the Sprague–Grundy function for positions with one pile made of n chips. (Please do not forget to prove correctness of your asnwer.)

A. Use the method of undetermined coefficients to find one solution of

y′′ − y′ + y = 4e^3t.

y(t)=

B. Find a particular solution to

y′′ − 2y′ + y = −16e^t.

y_p=

C. Find a particular solution to the differential equation

y′′ + 7y′ + 10y = 200t³.

y_p=

D. Find a particular solution to

y′′ + 6y′ + 5y = 20te^3t.

y_p=

E. Find the solution of

y′′ + 6y′ + 5y = 45e^0t

with y(0) = 5 and y′(0) = 1.

y =

if x1= Acos(wt+a) and x2+ Bcos(wt+b), how can I show that x1+x2=C cos (wt+c)? Furthermore how can I express the value of C and c? (using complex exponentials)

PROJECTIVE AND AFFINE GEOMETRY

¿ which is the affine application that transforms each line into another with its same direction and leaves at least two invariant points ?

Thank you for your explanations

List two practical reasons for using a small N design. Why is it usually important to return to baseline conditions in a small N design? Under what conditions would it not be desirable to return to baseline?

An individual has three umbrellas, some at his office, and some at home. If he is leaving home in the morning (or leaving work at night) and it is raining, he will take an umbrella, if there is one there. Otherwise, he gets wet. Assume that, independent of the past, it rains on each trip with probability 0.2. To formulate a Markov chain, let Xn be the number of umbrellas at his current location “before” he starts his n-th trip. Note that “current location” can either be home or office, depending on whether the trip is from home to office or vice versa. Find the transition probabilities of this Markov chain

True or false

A) you are trying to estimate a parameter of interest, you have two estimators with the same variance but the first estimator is an un-biased estimator and the second one is a biased estimator ,. In this case you should choose the un/ biased estimator

True?. False?
Please Explain

B.) you are trying to estimate Q1 ( the 25th percentile) for the anual household income in the us based on a sample of households earnings can you suggest an estimator for Q1!? Please explain why you believe this is a good estimator

What is the number of ordered sequences of length k where each digit is
taken from a set of size n?

What is the number of ordered sequences of length k where each digit is
taken from a set of size n without repetition?

What is the number of subsets of size k of a set of size n?