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Throughout, we use the following notation:

• 1n is the vector of all-ones of size n,

• 0n is the vector of all-zeros of size n,

• In is the n × n identity matrix.

Problem 1 (30 points) 6 + 10 + 14 = 30

Assume that z1, z2 and z3 are independent draws from N(1, 1), that is, a normal distribution

with mean and variance both equal to 1. Consider the vector

z =

z1 − z3

z2

.

(a) Fully specify the distribution of z.

(b) Can we choose a ∈ R

3

such that a

T z is independent of z1 − z2? If so specify one such

a, otherwise argue why not.

Let x = [x1, x2]

T be obtained by projecting z onto the span of the following set of vectors:

1

−1

,

−1

1

.

(c) Find the (joint) distribution of the random vector

z

x

.

1

Problem 2 (40 points) 8 + 8 + 8 + 16 = 40

Assume that the data y is generated according to the following model,

y = β1×1 + β2×2 + ε,

where y, ε, xj ∈ R

n and βj ∈ R for j = 1, 2. Assume that ε ∼ N(0, 2In) and

kx1k

2 = kx2k

2 = n, hx1, x2i = −

1

2

n,

and n = 10. Suppose that we mistakenly pose and fit the regression model y ∼ γx1. That

is, we compute the LSE estiamte

γb = argmin

γ∈R

ky − γx1k

2

and form the fitted-value vector µˆ = γbx1 and the residual vector e = y − γbx1.

Answer the following questions. Justify all your answers.

(a) For each of the following vectors, determine whether e is guaranteed to be orthogonal

to it or not:

x1, x2, µˆ, x1 + µˆ

(b) Is γb unbiased for estimating β1?

(c) Find the variance of γb.

(d) Find the joint distribution of

γb

e

.

2

Problem 3 (30 points) 6 + 12 + 12 = 30

Consider linear regression model y = Xβ + ε with transposed design matrix XT given by

t(X)

## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]

## [1,] 0 1 0 -1 -1 0 0 -1 0 0

## [2,] -1 0 1 0 0 -1 0 0 0 -1

## [3,] -1 1 0 -1 0 1 0 0 0 0

Note that the model does not include an intercept.

We fit a regression model with input y and design matrix X and obtain the following

estimate of the coefficient vector β:

coef(lm(y ~ 0 + X))

## X1 X2 X3

## -0.305 0.880 0.840

We also observe that kyk

2 = 5.462436. Answer the following questions:

(a) Find s

2

, the unbiased estimate of the noise variance.

(b) Test the null hypothesis 4β1 = β2 + β3 at level α = 0.05.

(c) What would the least-sqaures estimate of β be if we fit a regression model with response

vector y + x3 and design matrix X?

3

Compelling correspondence is essential to the achievement all things considered but since of the changing idea of the present working environments, successful correspondence turns out to be more troublesome, and because of the numerous impediments that will permit beneficiaries to acknowledge the plan of the sender It is restricted. Misguided judgments.In spite of the fact that correspondence inside the association is rarely completely open, numerous straightforward arrangements can be executed to advance the effect of these hindrances.

Concerning specific contextual analysis, two significant correspondence standards, correspondence channel determination and commotion are self-evident. This course presents the standards of correspondence, the act of general correspondence, and different speculations to all the more likely comprehend the correspondence exchanges experienced in regular daily existence. The standards and practices that you learn in this course give the premise to additionally learning and correspondence.

This course starts with an outline of the correspondence cycle, the method of reasoning and hypothesis. In resulting modules of the course, we will look at explicit use of relational connections in close to home and expert life. These incorporate relational correspondence, bunch correspondence and dynamic, authoritative correspondence in the work environment or relational correspondence. Rule of Business Communication In request to make correspondence viable, it is important to follow a few rules and standards. Seven of them are fundamental and applicable, and these are clear, finished, brief, obliging, right, thought to be, concrete. These standards are frequently called 7C for business correspondence. The subtleties of these correspondence standards are examined underneath: Politeness Principle: When conveying, we should build up a cordial relationship with every individual who sends data to us.

To be inviting and polite is indistinguishable, and politeness requires an insightful and amicable activity against others. Axioms are notable that gracious “pay of graciousness is the main thing to win everything”. Correspondence staff ought to consistently remember this. The accompanying standards may assist with improving courtesy:Preliminary considering correspondence with family All glad families have the mystery of progress. This achievement originates from a strong establishment of closeness and closeness. Indeed, through private correspondence these cozy family connections become all the more intently. Correspondence is the foundation of different affiliations, building solid partners of obedient devotion, improving family way of life, and assisting with accomplishing satisfaction (Gosche, p. 1). In any case, so as to keep up an amicable relationship, a few families experienced tumultuous encounters. Correspondence in the family is an intricate and alluring marvel. Correspondence between families isn’t restricted to single messages between families or verbal correspondence.

It is a unique cycle that oversees force, closeness and limits, cohesiveness and flexibility of route frameworks, and makes pictures, topics, stories, ceremonies, rules, jobs, making implications, making a feeling of family life An intelligent cycle that makes a model. This model has passed ages. Notwithstanding the view as a family and family automatic framework, one of the greatest exploration establishments in between family correspondence centers around a family correspondence model. Family correspondence model (FCP) hypothesis clarifies why families impart in their own specific manner dependent on one another ‘s psychological direction. Early FCP research established in media research is keen on how families handle broad communications data. Family correspondence was perceived as an exceptional scholastic exploration field by the National Communications Association in 1989. Family correspondence researchers were at first impacted by family research, social brain science, and relational hypothesis, before long built up the hypothesis and began research in a family framework zeroed in on a significant job. Until 2001, the primary issue of the Family Communication Research Journal, Family Communication Magazine, was given. Family correspondence is more than the field of correspondence analysts in the family. Examination on family correspondence is normally done by individuals in brain science, humanism, and family research, to give some examples models. However, as the popular family correspondence researcher Leslie Baxter stated, it is the focal point of this intelligent semantic creation measure making the grant of family correspondence special. In the field of in-home correspondence, correspondence is normally not founded on autonomous messages from one sender to one beneficiary, yet dependent on the dynamic interdependency of data shared among families It is conceptualized. The focal point of this methodology is on the shared trait of semantic development inside family frameworks. As such, producing doesn’t happen in vacuum, however it happens in a wide scope of ages and social exchange.

Standards are rules end up being followed when performing work to agree to a given objective. Hierarchical achievement relies significantly upon compelling correspondence. So as to successfully impart, it is important to follow a few standards and rules. Coming up next are rules to guarantee powerful correspondence: clearness: lucidity of data is a significant guideline of correspondence. For beneficiaries to know the message plainly, the messages ought to be sorted out in a basic language. To guarantee that beneficiaries can without much of a stretch comprehend the importance of the message, the sender needs to impart unmistakably and unhesitatingly so the beneficiary can plainly and unquestionably comprehend the data.>