Question

A. Which of the following models are linear? If a model is non-linear, indicate the non-linear term. I. y = ??(??x? + ??x?) + ? II. y = ??(x? ? x??) + ??(x? ? x??) + ? III. y = exp(?? + ??x) + ? IV. y = ?? + ??x?² + ??x?² + ? B. Write the corresponding mathematical models for the following linear models in "R": I. lm( y ~ x1 : x2 ) II. lm( y ~ I(1/x) + I(x^3) ) C. Let Y? ? N(2, 2) , Y? ? N(?4, 2) , Y? ? N(?2, 2) and Y? ? N(2, 4) be independent random variables. Answer the following questions and explain your workings. I. Find the distributions of X? = 2Y? + Y? and X? = Y? ? 2Y? II. Construct two independent random variables Z? = Z?(Y?, Y?, Y?, Y?) and Z? = Z?(Y?, Y?, Y?, Y?) such that Z? ? ?²? and Z? ? ?²? III. Construct a random variable W = W(Y?, Y?, Y?, Y?) such that W ? F?,? D. Briefly explain what are the standard assumptions on random errors in the simple linear regression model.

          A. Which of the following models are linear? If a model is non-linear, indicate the non-linear term.
I. y = ??(??x? + ??x?) + ?
II. y = ??(x? ? x??) + ??(x? ? x??) + ?
III. y = exp(?? + ??x) + ?
IV. y = ?? + ??x?² + ??x?² + ?

B. Write the corresponding mathematical models for the following linear models in "R":
I. lm( y ~ x1 : x2 )
II. lm( y ~ I(1/x) + I(x^3) )

C. Let Y? ? N(2, 2) , Y? ? N(?4, 2) , Y? ? N(?2, 2) and Y? ? N(2, 4) be independent random variables. Answer the following questions and explain your workings.
I. Find the distributions of X? = 2Y? + Y? and X? = Y? ? 2Y?
II. Construct two independent random variables Z? = Z?(Y?, Y?, Y?, Y?) and Z? = Z?(Y?, Y?, Y?, Y?) such that Z? ? ?²? and Z? ? ?²?
III. Construct a random variable W = W(Y?, Y?, Y?, Y?) such that W ? F?,?

D. Briefly explain what are the standard assumptions on random errors in the simple linear regression model.

A. Which of the following models are linear? If a model is non-linear, indicate the non-linear term.
I. y = ??(??x? + ??x?) + ?
II. y = ??(x? ? x??) + ??(x? ? x??) + ?
III. y = exp(?? + ??x) + ?
IV. y = ?? + ??x?² + ??x?² + ?

B. Write the corresponding mathematical models for the following linear models in "R":
I. lm( y x1 : x2 )
II. lm( y I(1/x) + I(x^3) )

C. Let Y? ? N(2, 2) , Y? ? N(?4, 2) , Y? ? N(?2, 2) and Y? ? N(2, 4) be independent random variables. Answer the following questions and explain your workings.
I. Find the distributions of X? = 2Y? + Y? and X? = Y? ? 2Y?
II. Construct two independent random variables Z? = Z?(Y?, Y?, Y?, Y?) and Z? = Z?(Y?, Y?, Y?, Y?) such that Z? ? ?²? and Z? ? ?²?
III. Construct a random variable W = W(Y?, Y?, Y?, Y?) such that W ? F?,?

D. Briefly explain what are the standard assumptions on random errors in the simple linear regression model.

Added by Christopher C.

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