Homework Answers
1.a)Here the R.V Y is obtained by shifting the origin of ε.So this is problem of Shift of origin. To find it's mean and standard deviation or variance we can easily say it by the property of origin shift of mean and variance.But the derivations are as follows:-
1.b)Calculating we get the sample mean=2.33
The sample standard deviation (with divisor n)=1.253
The Sample standard deviation (woth divisor n-1)=1.528
The calculations are:-
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la) If Y and ε are two variables,e-Normal(0, σ), and Y-Ao + AX + ε, where...
(20 points) Suppose X~N(25, 81). That is, X has a normal distribution with μ-25 and σ-81...
(20 points) Suppose X~N(25, 81). That is, X has a normal distribution with μ-25 and σ-81 la. Find a transformation of X that will give it a mean of zero and a variance of one (ie., standardize X lb. Find the probability that 18 < χ < 26. lc. Supposing Y10 +5X, find the mean of Y ld. Find the variance ofY
Let Xi, x,, ,X, be independent random variables with mean and variance σ . Let Y1-Y2, , Y, be independent random variables with mhean μ and variance a) Compute the expected value of W b) For what val...
Let Xi, x,, ,X, be independent random variables with mean and variance σ . Let Y1-Y2, , Y, be independent random variables with mhean μ and variance a) Compute the expected value of W b) For what value of a is the variance of W a minimum? σ: Let W-aX + (1-a) Y, where 0 < a < 1.
Let Xi, x,, ,X, be independent random variables with mean and variance σ . Let Y1-Y2, , Y, be independent random...
la) If variable Y-Normal(μ, σ), then in a sample of n observations, Judge TF of the...
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Σ (ax, + by, + c) O B. Ос. D. ax by nxc Consider the following...
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6. Let Z's be independent standard normal random variables. (a) Define X = Σ Z f...
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Onsider the process Y, = Y + Σ|e, where Yo ~ (μ, σ2) and the e's are 0-mean, a stationary process...
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onsider the process Y, = Y + Σ|e, where Yo ~ (μ, σ2) and the e's are 0-mean, a stationary process? independent identically distributed random variables with variance 1. Is (Y How about the process ▽Yǐ = Yt-)t-1 ? Explain.
Suppose X and Y are random variables such that fY (y|X = x) has a normal...
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5. Suppose X and Y are random variables such that E(X)=E(Y) = θ, Var(X) = σ...
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The random vector Y = (Y1, ..., Yn)T is such that Y = Xβ + ε,...
The random vector Y = (Y1, ...,
Yn)T is such that Y = Xβ + ε, where X is an n
× p full-rank matrix of known constants, β is a p-length vector of
unknown parameters, and ε is an n-length vector of random
variables. A multiple linear regression model is fitted to the
data.
(a) Write down the multiple linear regression model assumptions in
matrix format.
(b) Derive the least squares estimator β^ of β.
(c) Using the data:...
4. Suppose we run a regression model Y = β0+AX+U when the true model is Y-a0+ α1X2 + V. Assume th...
a,b,c,d
4. Suppose we run a regression model Y = β0+AX+U when the true model is Y-a0+ α1X2 + V. Assume that the true model satisfies all five standard assumptions of a simple regression model discussed in class. (a) Does the regression model we are running satisfy the zero conditional mean assumption? (b) Find the expected value of A (given X values). (e) Does the regression model we are running satisfy homoscedasticity? d) Find the variance of pi (given X...
(20 points) Suppose X~N(25, 81). That is, X has a normal distribution with μ-25 and σ-81 la. Find a transformation of X that will give it a mean of zero and a variance of one (ie., standardize X lb. Find the probability that 18 < χ < 26. lc. Supposing Y10 +5X, find the mean of Y ld. Find the variance ofY
Let Xi, x,, ,X, be independent random variables with mean and variance σ . Let Y1-Y2, , Y, be independent random variables with mhean μ and variance a) Compute the expected value of W b) For what value of a is the variance of W a minimum? σ: Let W-aX + (1-a) Y, where 0 < a < 1.
Let Xi, x,, ,X, be independent random variables with mean and variance σ . Let Y1-Y2, , Y, be independent random...
la) If variable Y-Normal(μ, σ), then in a sample of n observations, Judge TF of the following statements on the sample mean Y A) (Y-μ)/σ ~ Normal (μ, σ)-- B) (F-H)/Normal (0, 1) C) (Ỹ-p)/(vn) ~ Normal(0.1)- D) (Y-μ)/( ) ~ t(n-1) lb) Which of the following statements in a hypothesis test defines the i) Type I error Type II error iii) Power A) Pr(Reject Ho) B) Pr(Do not reject Ho) C) Pr(Ho is true) D) Pr(Ho is false) E)...
Σ (ax, + by, + c) O B. Ос. D. ax by nxc Consider the following linear transformation of a random variable where ux is the mean of x and o, is the standard deviation. Then the expected value and the standard deviation of Y are given as: and σχ. O B. 0 and 1 O c. cannot be computed because Y is not a linear function ofx. O D. 1 and 1
6. Let Z's be independent standard normal random variables. (a) Define X = Σ Z f X. (b) Define Y = 4 Σ zi. Find the mean and variance of Y. (Hint: Use the fact E(Z Z,)-0 for any i fj, i,j 1,2,3,4.) i. Find the mean and variance o i=1 4 i=1
onsider the process Y, = Y + Σ|e, where Yo ~ (μ, σ2) and the e's are 0-mean, a stationary process? independent identically distributed random variables with variance 1. Is (Y How about the process ▽Yǐ = Yt-)t-1 ? Explain.
onsider the process Y, = Y + Σ|e, where Yo ~ (μ, σ2) and the e's are 0-mean, a stationary process? independent identically distributed random variables with variance 1. Is (Y How about the process ▽Yǐ = Yt-)t-1 ? Explain.
5. Suppose X and Y are random variables such that E(X)=E(Y) = θ, Var(X) = σ and Var(Y)-吆 . Consider a new random variable W = aX + (1-a)Y (a) Show that W is unbiased for θ. (b) If X and Y are independent, how should the constant a be chosen in order to minimize the variance of W?
The random vector Y = (Y1, ...,
Yn)T is such that Y = Xβ + ε, where X is an n
× p full-rank matrix of known constants, β is a p-length vector of
unknown parameters, and ε is an n-length vector of random
variables. A multiple linear regression model is fitted to the
data.
(a) Write down the multiple linear regression model assumptions in
matrix format.
(b) Derive the least squares estimator β^ of β.
(c) Using the data:...
a,b,c,d
4. Suppose we run a regression model Y = β0+AX+U when the true model is Y-a0+ α1X2 + V. Assume that the true model satisfies all five standard assumptions of a simple regression model discussed in class. (a) Does the regression model we are running satisfy the zero conditional mean assumption? (b) Find the expected value of A (given X values). (e) Does the regression model we are running satisfy homoscedasticity? d) Find the variance of pi (given X...
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