Suppose that, for n observations y1,,yn, we have specified a multiple linear regression model of
the form yi=0+1xi1+2xi2+ei where the ei are uncorrelated errors, with E(eiX)=0 and Var(eiX)=2.
Consider the situation where we are interested not in the values of the coefficients 1 and 2
separately, but instead in their difference =12. a) Show that the estimator ^=^1^2 is an umbiased
estimator of . That is, show that E(^X)= b) Derive an expression for the variance of this estimator,
Var(), in terms of the relevant entries of the variance-covariance matrix Var(), where is the OLS
estimator of =(0,1,2)T..
Suppose that you wished to copy all of the text files from y.pdfactivecomputers1
Suppose that you wished to copy all of the text files from your ~/UnixCourse/fileAsst directory and
all of the files from your ~/UnixCourse/fileAsst/Planes directory into your current directory. (without
typing out the name of each individual file). What command would you give to make that copy?
the current directory is ~/UnixCourse/commandsAsst
should be a linux cp command.
suppose that XY and Z are yointly oirtmiutedirandom yariab.pdfactivecomputers1
suppose that X,Y, and Z are yointly oirtmiutedirandom yariables, that is, they are detined on the
same sample space. suppose that we also have the following. E(X)=2Var(X)=2E(Y)=6Var(Y)=21E(
z)=0Var(z)=18 Compute the values of the expressions below E(52Z)=5(23y+4X)=var(3Z)5=E(x2)=.
Suppose that XY is uniformly distributed over the region .pdfactivecomputers1
Suppose that (X,Y) is uniformly distributed over the region {(x,y):0<y<x<1} Find: a) the joint density
of (X,Y); b) the marginal densities fX(x) and fY(y). c) Are X and Y independent? d) Find E(X) and E
(Y)..
Suppose that XY and Z are jointly distributed random varia.pdfactivecomputers1
Suppose that X,Y, and Z are jointly distributed random variables, that is, they are defined on the
same sample space. Suppose that we also have the following. E(X)=6Var(X)=11E(Y)=0Var(Y)=42
E(Z)=1Var(Z)=2 Compute the values of the expressions below. E(1Z)=E(45X2Z)=Var(25X)=E(4X2
)=.
Suppose that XUnif01 You draw a sample of X and observe .pdfactivecomputers1
Suppose that XUnif(0,1). You draw a sample of X and observe the value x. Then you draw a
sample of (YX=x)Unif(x,1). What is the expected value of Y ? 3/4 1/2 2(1+x)) 1/4.
Suppose that X1X2Xa is a random sample of size n is to b.pdfactivecomputers1
Suppose that X1,X2,,Xa is a random sample of size n is to be taken from a geometric population
(# of trials) with success probability p. The MLE of p is 1/X. Using the delta method, determine the
approximate distribution of 1/X. 1/X is approximately normal with mean and variance.
Suppose that XndX and that Ynp0 Show that nlimPXn+YnzFXz.pdfactivecomputers1
Suppose that XndX and that Ynp0. Show that nlimP(Xn+Ynz)FX(z) for >0 and any z. Justify every
step you take. If you fail to give the correct justification for a step, no credit will be given..
Suppose that we have two independent random samples X1X2.pdfactivecomputers1
Suppose that we have two independent random samples: X1,X2,,Xn are exponential (), and Y1,Y2
,,Ym are exponential (). (a) Find the LRT of H0:=, versus H1:=. (b) Show that the test in part (a)
can be based on the statistic T=i=1nXi+j=1mYji=1nXi..
Suppose that you wished to copy all of the text files from y.pdfactivecomputers1
Suppose that you wished to copy all of the text files from your ~/UnixCourse/fileAsst directory and
all of the files from your ~/UnixCourse/fileAsst/Planes directory into your current directory. (without
typing out the name of each individual file). What command would you give to make that copy?
the current directory is ~/UnixCourse/commandsAsst
should be a linux cp command.
suppose that XY and Z are yointly oirtmiutedirandom yariab.pdfactivecomputers1
suppose that X,Y, and Z are yointly oirtmiutedirandom yariables, that is, they are detined on the
same sample space. suppose that we also have the following. E(X)=2Var(X)=2E(Y)=6Var(Y)=21E(
z)=0Var(z)=18 Compute the values of the expressions below E(52Z)=5(23y+4X)=var(3Z)5=E(x2)=.
Suppose that XY is uniformly distributed over the region .pdfactivecomputers1
Suppose that (X,Y) is uniformly distributed over the region {(x,y):0<y<x<1} Find: a) the joint density
of (X,Y); b) the marginal densities fX(x) and fY(y). c) Are X and Y independent? d) Find E(X) and E
(Y)..
Suppose that XY and Z are jointly distributed random varia.pdfactivecomputers1
Suppose that X,Y, and Z are jointly distributed random variables, that is, they are defined on the
same sample space. Suppose that we also have the following. E(X)=6Var(X)=11E(Y)=0Var(Y)=42
E(Z)=1Var(Z)=2 Compute the values of the expressions below. E(1Z)=E(45X2Z)=Var(25X)=E(4X2
)=.
Suppose that XUnif01 You draw a sample of X and observe .pdfactivecomputers1
Suppose that XUnif(0,1). You draw a sample of X and observe the value x. Then you draw a
sample of (YX=x)Unif(x,1). What is the expected value of Y ? 3/4 1/2 2(1+x)) 1/4.
Suppose that X1X2Xa is a random sample of size n is to b.pdfactivecomputers1
Suppose that X1,X2,,Xa is a random sample of size n is to be taken from a geometric population
(# of trials) with success probability p. The MLE of p is 1/X. Using the delta method, determine the
approximate distribution of 1/X. 1/X is approximately normal with mean and variance.
Suppose that XndX and that Ynp0 Show that nlimPXn+YnzFXz.pdfactivecomputers1
Suppose that XndX and that Ynp0. Show that nlimP(Xn+Ynz)FX(z) for >0 and any z. Justify every
step you take. If you fail to give the correct justification for a step, no credit will be given..
Suppose that we have two independent random samples X1X2.pdfactivecomputers1
Suppose that we have two independent random samples: X1,X2,,Xn are exponential (), and Y1,Y2
,,Ym are exponential (). (a) Find the LRT of H0:=, versus H1:=. (b) Show that the test in part (a)
can be based on the statistic T=i=1nXi+j=1mYji=1nXi..
Suppose that the profit of a portfolio of assets X follows.pdfactivecomputers1
Suppose that the profit of a portfolio of assets (X) follows a uniform distribution over the interval (2,
2).
(i) Write down the equation and plot the CDF of X.
(ii) Calculate the 1% VaR of the profit X.
(iii) Calculate the expected shortfall at 1% level of the profit X.
(iv) Calculate the tail conditional expectation at 1% level of the profit X..
Suppose that the TOEFL Writing score follows a Normal distri.pdfactivecomputers1
Suppose that the TOEFL Writing score follows a Normal distribution with an average of 15 and a
standard deviation of 2. A random sample of eight TOEFL test-takers is drawn. a) Provide a
description of the most appropriate statistic. [2] b) Identify the sampling distribution of answer you
have in part (a). [4] c) What is the probability that the average TOEFL Writing score of the eight
randomly selected TOEFL test-takers.
Suppose that the US dollarpound sterling exchange rate eq.pdfactivecomputers1
Suppose that the U.S. dollar-pound sterling exchange rate equals to $1.30/f, while the 3-month
forward rate is $1.40/. The yield on 1-year U.S. and U.K. Treasury bills are 8% and 12%,
respectively:
Calculate the 3-month covered interest differential using the exact formula (f = (i-i*)/(1+
i*)). Are there net gains to be made?.
Suppose that the multiple linear regression model yixiT+ei .pdfactivecomputers1
Suppose that the multiple linear regression model yi=xiT+ei holds for i=1,,n, where the ei are
independent errors, with E(eiX=xi)=0 but, unlike the standard setting, with different variances Var(
eiX=xi)=i2. a) Show that the OLS estimate ^=(XTX)1XTY is still an unbiased estimate of . That is,
show that E[^X]= under the model assumptions. b) Write the variance matrix of the errors ei as a
diagonal matrix, say , with diagonal elements (12,,n2). Show that Var(^)=(XTX)1XTX(XTX)1..
Suppose that the marginal propensity to save is dyd5057y+8.pdfactivecomputers1
Suppose that the marginal propensity to save is dyd5=0.57y+81(inbillionsofdollars) and that
consumption is $1 billion when disposable income is $0. Find the national consumption function.
(Round the constant of integration to the fearest hundredth.) C(y)=.
Suppose that the heights of adult women in the United States.pdfactivecomputers1
Suppose that the heights of adult women in the United States are normally distributed with a mean
of 65 inches and a standard deviation of 2.4 inches. Martina is taller than 80% of the population of
U.S. women. How tall (in inches) is Martina? Carry your intermediate computations to at least four
decimal places. Round your answer to one decimal place..
Suppose that the annual rainfall in Ferndale California is.pdfactivecomputers1
Suppose that the annual rainfall in Ferndale, California, is known to be normally distributed, with a
mean of 35.5 inches and a standard deviation of 2.5 inches. What is the probability that the rainfall
in a given year will exceed 30.5 inches in Ferndale? (You may need to use the standard normal
distribution table. Round your answer to three decimal places.)
1.
Suppose that the annual rate of return for a common biotechn.pdfactivecomputers1
Suppose that the annual rate of return for a common biotechnology stock is normally distributed
with a mean of 4% and a standard deviation of 5%. Find the probability that the one-year return of
this stock will be negative. Round your answer to at least four decimal places..
Suppose that during the course of human history whales hav.pdfactivecomputers1
Suppose that, during the course of human history, whales have evolved to be better able to avoid
human hunters. Given what you know about evolution, which one of these things likely played the
biggest role in contributing to this evolutionary process? Whales that happened to have genes that
allowed them to stay underwater longer survived. These whales then passed those genes on to
their offspring. Some whales were better at having offspring than other whales. This caused them
to stay underwater longer (where they were harder for people to find). As whales were being
hunted, some of them underwent mutations that allowed them to stay underwater longer (where
they were harder for people to find)..
Suppose that temperatures in C of the last 10 days in Tab.pdfactivecomputers1
Suppose that temperatures (in C ) of the last 10 days in Tabuk were: 21,23,27,19,25,21,26,31,29
and 24 a) Find the 17th percentile (P17) of the temperatures, what does this value mean? 10)
Suppose that the mean of a set of 20 observations is 70 . If the mean of the first 5 observations in
this set is 100 , what is the mean of the last 15 observations?.
Suppose that an individual has a Utility function represente.pdfactivecomputers1
Suppose that an individual has a Utility function represented by a CES function. The utility function
of the individual is given as: U(x,y)=x1/2+y1/2 c. Is the demand more elastic or inelastic than a
Cobb-Douglas Utility function? Use the Slutsky matrix to illustrate..
Suppose that an airfine uses a seat with of 168 in Assume .pdfactivecomputers1
Suppose that an airfine uses a seat with of 16.8 in. Assume men have hip benactis that are
nonmelly dstrbuted wath a mean of 14 in in and a standard deriation of t.1 in Corpiche partt (a)
through (e) beiow. (a) Find the probability that if an individual man is randomly sebcted, his hip
breadth will be greater than 16.8 in. The prebabity is (Round to four decirnal piaces as needed).
Suppose that Jill Leonard Tavello the Vice President of Cul.pdfactivecomputers1
Suppose that Jill Leonard Tavello, the Vice President of Culture and Communication for Stew
Leonard's, wanted to track the effectiveness of the fufure Leader Program at Stew Leonard's.
Which of the following metrics would be effective for this analysis? Check all that apply. Annual
cost of the program Success rates of individuals promoted Organizational profitability Tumover
rates for individuals that go through the program According to the Nine-Box Talent Grid, which one
of the following employees is most likly to get put into the Ficure Leader Program at 5 tew
Leonard's? A future star A consistent star A current star A key player.
Suppose that 42 of primary care doctors think their patient.pdfactivecomputers1
Suppose that 42% of primary care doctors think their patients receive unnecessary medical care.
A sample of 300 primary care doctors were taken. a. Show the sampling distribution of the
proportion of the doctors who think their patients receive unnecessary medical care. b. What is the
probability that the sample proportion will be within 0.03 of the population proportion?.
Suppose N gt 0 firms make a simultaneous and onetime entr.pdfactivecomputers1
Suppose N > 0 firms make a simultaneous and one-time entry decision whether to enter a new
market. Entry requires
expending a sunk cost E 0, and entering firms produce homogeneous products. If only one firm
enters, then it is a
monopolist that earns profit M > 0, not counting the entry cost. If more than one firm enters, then
severe price
competition leads each entering firm to earn 0 profit, not counting the entry cost. Assume that E <
M
, so that entry can possibly be profitable.
Specify a firms probability of entry in a Nash equilibrium of the game in which all firms enter with
the
same probability.
Suppose new data on GDP is released It tells us that GDP gr.pdfactivecomputers1
Suppose new data on GDP is released. It tells us that GDP grew quickly, and the increase was
driven by a large increase in inventory. What does this tell us about the future? GDP is likely to
rise because companies will increase production GDP is likely to fall because companies will
reduce production The answer is not clear - high inventories could cause GDP to rise or fall.
Suppose Margie is engaged to Dennis on April 16 2025 Denn.pdfactivecomputers1
Suppose Margie is engaged to Dennis on April 16, 2025. Dennis wants to accumulate funds for
retirement. He will save $300 per month for the 30 years following their marriage. Assuming
monthly compounding. If he earns a constant return of 8%.
Now suppose that Dennis is unlucky and earns a -16% return for the first five years of his
investment. Then the market rebounds and produces 12.8% for the remaining 25 years of his
investment.
What was his arithmetic average rate of return on his investment in this case?.
Suppose Leah withdraws 7500 from her moneymarket mutual f.pdfactivecomputers1
Suppose Leah withdraws $7,500 from her money-market mutual fund and deposits this money
into her son's chequing account at a chartered bank. Then
A.both M2 and M2+ will increase by $7,500.
B.the values of M2 and M2+ will remain unchanged.
C.M2 will decrease by $7,500, but M2+ will increase by $7,500.
D.M2 will increase by $7,500, and the value of M2+ will stay the same.
E.both M2 and M2+ will decrease by $7,500..
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Suppose that the profit of a portfolio of assets X follows.pdfactivecomputers1
Suppose that the profit of a portfolio of assets (X) follows a uniform distribution over the interval (2,
2).
(i) Write down the equation and plot the CDF of X.
(ii) Calculate the 1% VaR of the profit X.
(iii) Calculate the expected shortfall at 1% level of the profit X.
(iv) Calculate the tail conditional expectation at 1% level of the profit X..
Suppose that the TOEFL Writing score follows a Normal distri.pdfactivecomputers1
Suppose that the TOEFL Writing score follows a Normal distribution with an average of 15 and a
standard deviation of 2. A random sample of eight TOEFL test-takers is drawn. a) Provide a
description of the most appropriate statistic. [2] b) Identify the sampling distribution of answer you
have in part (a). [4] c) What is the probability that the average TOEFL Writing score of the eight
randomly selected TOEFL test-takers.
Suppose that the US dollarpound sterling exchange rate eq.pdfactivecomputers1
Suppose that the U.S. dollar-pound sterling exchange rate equals to $1.30/f, while the 3-month
forward rate is $1.40/. The yield on 1-year U.S. and U.K. Treasury bills are 8% and 12%,
respectively:
Calculate the 3-month covered interest differential using the exact formula (f = (i-i*)/(1+
i*)). Are there net gains to be made?.
Suppose that the multiple linear regression model yixiT+ei .pdfactivecomputers1
Suppose that the multiple linear regression model yi=xiT+ei holds for i=1,,n, where the ei are
independent errors, with E(eiX=xi)=0 but, unlike the standard setting, with different variances Var(
eiX=xi)=i2. a) Show that the OLS estimate ^=(XTX)1XTY is still an unbiased estimate of . That is,
show that E[^X]= under the model assumptions. b) Write the variance matrix of the errors ei as a
diagonal matrix, say , with diagonal elements (12,,n2). Show that Var(^)=(XTX)1XTX(XTX)1..
Suppose that the marginal propensity to save is dyd5057y+8.pdfactivecomputers1
Suppose that the marginal propensity to save is dyd5=0.57y+81(inbillionsofdollars) and that
consumption is $1 billion when disposable income is $0. Find the national consumption function.
(Round the constant of integration to the fearest hundredth.) C(y)=.
Suppose that the heights of adult women in the United States.pdfactivecomputers1
Suppose that the heights of adult women in the United States are normally distributed with a mean
of 65 inches and a standard deviation of 2.4 inches. Martina is taller than 80% of the population of
U.S. women. How tall (in inches) is Martina? Carry your intermediate computations to at least four
decimal places. Round your answer to one decimal place..
Suppose that the annual rainfall in Ferndale California is.pdfactivecomputers1
Suppose that the annual rainfall in Ferndale, California, is known to be normally distributed, with a
mean of 35.5 inches and a standard deviation of 2.5 inches. What is the probability that the rainfall
in a given year will exceed 30.5 inches in Ferndale? (You may need to use the standard normal
distribution table. Round your answer to three decimal places.)
1.
Suppose that the annual rate of return for a common biotechn.pdfactivecomputers1
Suppose that the annual rate of return for a common biotechnology stock is normally distributed
with a mean of 4% and a standard deviation of 5%. Find the probability that the one-year return of
this stock will be negative. Round your answer to at least four decimal places..
Suppose that during the course of human history whales hav.pdfactivecomputers1
Suppose that, during the course of human history, whales have evolved to be better able to avoid
human hunters. Given what you know about evolution, which one of these things likely played the
biggest role in contributing to this evolutionary process? Whales that happened to have genes that
allowed them to stay underwater longer survived. These whales then passed those genes on to
their offspring. Some whales were better at having offspring than other whales. This caused them
to stay underwater longer (where they were harder for people to find). As whales were being
hunted, some of them underwent mutations that allowed them to stay underwater longer (where
they were harder for people to find)..
Suppose that temperatures in C of the last 10 days in Tab.pdfactivecomputers1
Suppose that temperatures (in C ) of the last 10 days in Tabuk were: 21,23,27,19,25,21,26,31,29
and 24 a) Find the 17th percentile (P17) of the temperatures, what does this value mean? 10)
Suppose that the mean of a set of 20 observations is 70 . If the mean of the first 5 observations in
this set is 100 , what is the mean of the last 15 observations?.
Suppose that an individual has a Utility function represente.pdfactivecomputers1
Suppose that an individual has a Utility function represented by a CES function. The utility function
of the individual is given as: U(x,y)=x1/2+y1/2 c. Is the demand more elastic or inelastic than a
Cobb-Douglas Utility function? Use the Slutsky matrix to illustrate..
Suppose that an airfine uses a seat with of 168 in Assume .pdfactivecomputers1
Suppose that an airfine uses a seat with of 16.8 in. Assume men have hip benactis that are
nonmelly dstrbuted wath a mean of 14 in in and a standard deriation of t.1 in Corpiche partt (a)
through (e) beiow. (a) Find the probability that if an individual man is randomly sebcted, his hip
breadth will be greater than 16.8 in. The prebabity is (Round to four decirnal piaces as needed).
Suppose that Jill Leonard Tavello the Vice President of Cul.pdfactivecomputers1
Suppose that Jill Leonard Tavello, the Vice President of Culture and Communication for Stew
Leonard's, wanted to track the effectiveness of the fufure Leader Program at Stew Leonard's.
Which of the following metrics would be effective for this analysis? Check all that apply. Annual
cost of the program Success rates of individuals promoted Organizational profitability Tumover
rates for individuals that go through the program According to the Nine-Box Talent Grid, which one
of the following employees is most likly to get put into the Ficure Leader Program at 5 tew
Leonard's? A future star A consistent star A current star A key player.
Suppose that 42 of primary care doctors think their patient.pdfactivecomputers1
Suppose that 42% of primary care doctors think their patients receive unnecessary medical care.
A sample of 300 primary care doctors were taken. a. Show the sampling distribution of the
proportion of the doctors who think their patients receive unnecessary medical care. b. What is the
probability that the sample proportion will be within 0.03 of the population proportion?.
Suppose N gt 0 firms make a simultaneous and onetime entr.pdfactivecomputers1
Suppose N > 0 firms make a simultaneous and one-time entry decision whether to enter a new
market. Entry requires
expending a sunk cost E 0, and entering firms produce homogeneous products. If only one firm
enters, then it is a
monopolist that earns profit M > 0, not counting the entry cost. If more than one firm enters, then
severe price
competition leads each entering firm to earn 0 profit, not counting the entry cost. Assume that E <
M
, so that entry can possibly be profitable.
Specify a firms probability of entry in a Nash equilibrium of the game in which all firms enter with
the
same probability.
Suppose new data on GDP is released It tells us that GDP gr.pdfactivecomputers1
Suppose new data on GDP is released. It tells us that GDP grew quickly, and the increase was
driven by a large increase in inventory. What does this tell us about the future? GDP is likely to
rise because companies will increase production GDP is likely to fall because companies will
reduce production The answer is not clear - high inventories could cause GDP to rise or fall.
Suppose Margie is engaged to Dennis on April 16 2025 Denn.pdfactivecomputers1
Suppose Margie is engaged to Dennis on April 16, 2025. Dennis wants to accumulate funds for
retirement. He will save $300 per month for the 30 years following their marriage. Assuming
monthly compounding. If he earns a constant return of 8%.
Now suppose that Dennis is unlucky and earns a -16% return for the first five years of his
investment. Then the market rebounds and produces 12.8% for the remaining 25 years of his
investment.
What was his arithmetic average rate of return on his investment in this case?.
Suppose Leah withdraws 7500 from her moneymarket mutual f.pdfactivecomputers1
Suppose Leah withdraws $7,500 from her money-market mutual fund and deposits this money
into her son's chequing account at a chartered bank. Then
A.both M2 and M2+ will increase by $7,500.
B.the values of M2 and M2+ will remain unchanged.
C.M2 will decrease by $7,500, but M2+ will increase by $7,500.
D.M2 will increase by $7,500, and the value of M2+ will stay the same.
E.both M2 and M2+ will decrease by $7,500..
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Leveraging Generative AI to Drive Nonprofit Innovation
Suppose that for n observations y1yn we have specified .pdf
1. Suppose that, for n observations y1,,yn, we have specified a multiple linear regression model of
the form yi=0+1xi1+2xi2+ei where the ei are uncorrelated errors, with E(eiX)=0 and Var(eiX)=2.
Consider the situation where we are interested not in the values of the coefficients 1 and 2
separately, but instead in their difference =12. a) Show that the estimator ^=^1^2 is an umbiased
estimator of . That is, show that E(^X)= b) Derive an expression for the variance of this estimator,
Var(), in terms of the relevant entries of the variance-covariance matrix Var(), where is the OLS
estimator of =(0,1,2)T.
