MGEB11H3 Lecture Notes - Lecture 8: Random Variable, Sample Space, All The Same

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amberant375

3 Nov 2018

School

UTSC

Department

Economics for Management Studies

Course

MGEB11H3

Professor

Victor Yu

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Document Summary

Mgeb11 lecture 8 chapter 6: continuous random variables: uniform distribution, example: consider the following graph. Let x be a random variable, 20 < x < 60. F(x) = 1/40, 20 < x < 60 we say x is uniform from 20 to 60. In general, let x be a random variable. X is a uniform randomvariable from a to b: let x be uniform from 10 to 50, find f(x) Answer: f(x) = 1/40: p (15= 32) = area to the right of 32. = 18/40 = 9/20: p (x = 32) = area from 32 to 32. = 0 (no line: p(15 < x < 32) = p (15<= x <= 32) Let x be uniform from a to b. Example: in previous example: the mean of x is, the variance of x is.

Related Questions

1. You are given only three quarterly seasonal indices and quarterly seasonally adjusted data for the entire year. What is the raw data value for Q4? Raw data is not adjusted for seasonality.

Quarter Seasonal Index Seasonally Adjusted Data

Q1 .80 295

Q2 .85 299

Q3 1.15 270

Q4 --- 271

2. One model of exponential smoothing will provide almost the same forecast as a liner trend method. What are linear trend intercept and slope counterparts for exponential smoothing?

A. Alpha and Delta

B. Delta and Gamma

C. Alpha and Gamma

D. Standard Deviation and Mean

3. When performing correlation analysis what is the null hypothesis? What measure in Minitab is used to test it and to be 95% confident in the significance of correlation coefficient.

A. Ho: r = .05 p < .5

B. Ho: r = 0 p >.05

C. Ho: r ? 0 p?.05

D. Ho: r = 0 p?.05

In decomposition what does the cycle factor (CF) of .80 represent for a monthly forecast estimate of a Y variable?

A. The estimated value is 80% of the average monthly seasonal estimate.
B. The estimate is .80 of the forecasted Y trend value.
C. The estimated value is .80 of the historical average CMA values.
D. The estimated value has 20% more variation than the average historical Y data values.

5. A Wendy's franchise owner notes that the sales per store has fallen below the stated national Wendy's outlet average of $1,368,000. He asserts a change has occurred that reduced the fast food eating habits of Americans. What is his hypothesis (H1) and what type of test for significance must be applied?

A. H1: u ? $1,368,000 A one-tailed t-test to the left.
B. H1: u = $1,368,000 A two-tailed t-test.
C. H1: u < $1,368,000 A one-tailed t-test to the left.
D. H1: p < $1,368,000 A one-tailed test to the right

6. As the sample size from a population increases, for a given level of significance what happens to the null hypothesis rejection region and size of the t-table value?

A. The rejection region and the t-table value generally gets smaller for sample size below 31.
B. The rejection region gets larger and the t-table value generally gets smaller for sample sizes below 31.
C. The rejection region remains unchanged while the t-table value gets smaller for all sample sizes.
D. The zero mean hypothesis region gets larger and the t-table value gets larger as well for sample sizes below 31.

7. You obtained autocorrelation LBQ value of 18.58 for the 12^th lag from a data sample. Are the data significantly correlated at the lag examined or not? You want to be 95% confident in your answer.

A. Yes. The data are significantly correlated through the 12th lag.
B. No. The data are not significantly correlated through the 12^th lag.

C. No. Only the 12 lag period is not correlated.

D. You cannot tell since the number of sample observations is not provided.

E. The p-value is above .05 so the data is correlated.

8. Sometimes forecasters get lazy or forgetful and do not check the significance of XY data correlations and use the X variable to forecast Y. What is the result of this?

A. Type 2 error
B. Autocorrelation error
C. Type 3 error
D. Type 1 error

9. Do error measures indicate the statistical significance of forecast model variables?

A. Yes. They move in the same direction as statistical significance.
B. Yes. As error measures decrease the variable significance increases.
C. No They indicate only the magnitude of estimate error.
D. No. They indicate only the statistical significance of a forecast or fitted values.

10. In exponential smoothing what is the weight of the alpha coefficient for a time series data observation from the 3^rd previous period if the original alpha value is set at .8?

A. The weight cannot be calculated since the data observation is not given.
B. The weight is zero since the alpha value is set relatively high.
C. .548
D. .0064

11.Given the data series below for variables Y (Monthly Inventory Balance) and X (Monthly Sales) are they significantly correlated at the 95% confidence level and how can you tell?

Ending Inv. Bal. Y		Monthly Sales X
1544		5053
1913		5052
2028		7507
1178		2887
1554		3880
1910		4454
1208		3855
2467		8824
2101		5716

A. Yes. The correlation coefficient is .873 that is greater than .05.
B. Yes. The correlation p-value is .002 which is less than .05.
C. No. The correlation coefficient is above the p-value.
D.No. The correlation p-value is greater than the 95% confidence level.

12. You have forecast the sales for your company for the last 12 months and the forecast residuals are shown below. Are these residuals to be considered random?
Residuals

-24

-348

-892

-62

-378

-489

-342

490

578

198

A. Yes, since the residuals randomly vary in magnitude.
B. Yes since the residuals are positive and negative and vary in magnitude.
C. No, since the residuals are stationary and vary in magnitude.
D. No, since the residuals indicate positive slope.

13. Given the residuals in problem 12 above what is the RMSE for the forecast?

A. -101.0
B. 411.8
C. 169603
D. 220.1

14. Which form of exponential smoothing can result in a na

Suppose that, instead of taking this test, you could either have worked and earned income or partied and had a pleasurable time. Your opportunity cost of taking the test is the

	a.	forgone working or partying, depending on which was your next best choice.
	b.	forgone work.
	c.	forgone working and partying.
	d.	test because you are taking it.

1 points

QUESTION 2

A trend is

	a.	a measure of closeness on a scatter diagram.
	b.	a general tendency for a variable to rise or fall.
	c.	the maximum value of a variable.
	d.	the difference between the maximum value of a variable and the minimum value of the variable.

1 points

QUESTION 3

A time-series graph measures

	a.	time on the y-axis and the variable or variables in which we are interested on the x-axis.
	b.	time on the x-axis and the variable or variables in which we are interested on the y-axis.
	c.	the value of one variable against the value of another variable.
	d.	the value of an economic variable for different groups in a population at a point in time.

1 points

QUESTION 4

Canada has nationalized health care, so that everyone, regardless of their ability to pay, has some access to health care. Based on this observation, Canada has decided that "everyone, regardless of their ability to pay" is the answer to what microeconomic question?

	a.	Why will we offer health care?
	b.	How will health care be produced?
	c.	For whom will health care be produced?
	d.	What type of health care will be produced and in what quantity?

1 points

QUESTION 5

As a shoe factory adds more workers, shoe production grows, reaches a maximum, and then shrinks. In a diagram that has the number of workers on the horizontal axis and the number of shoes on the vertical axis, the relationship between the number of workers and the number of shoes starts as ________ and then, after the maximum point, is ________.

	a.	positive; negative
	b.	negative; positive
	c.	linear; negative
	d.	positive; linear

1 points

QUESTION 6

Microeconomics includes the study of the

	a.	choices made by individuals and businesses.
	b.	recessions and inflation in the global economy.
	c.	aggregate effects on the national economy.
	d.	reasons why the government changes interest rates.

1 points

QUESTION 7

When Fresh Express Salads decides to mechanically pick all of its lettuce, it directly answers the ________ question.

	a.	when
	b.	for whom
	c.	how
	d.	what

1 points

QUESTION 8

Proponents of cuts in income tax rates argue that when income tax rates are cut, workers have an incentive to increase their work hours. This argument is based on the assumption that

	a.	the opportunity cost of working is negative.
	b.	the marginal cost of each additional work hour is not important to most workers.
	c.	workers are irrational.
	d.	workers make decisions based on the marginal benefit of each hour worked compared to the marginal cost of work.

1 points

QUESTION 9

A choice made by comparing all relevant alternatives systematically and incrementally is

	a.	an opportunity cost.
	b.	a benefit.
	c.	a sunk cost.
	d.	a choice on the margin.

1 points

QUESTION 10

A normative statement

i. can be tested as to whether it is true or false.

ii. is considered negative.

iii. depends on a person's values.

	a.	i, ii, and iii
	b.	iii only
	c.	i and iii
	d.	ii and iii

1 points

QUESTION 11

Normative statements

i. describe how the world is.

ii. describe how the world ought to be.

iii. depend on people's values and cannot be tested.

	a.	iii only.
	b.	i only.
	c.	ii and iii.
	d.	ii only.

1 points

QUESTION 12

In order to determine whether to major in economics, a rational individual compares the ________ of the decision.

	a.	positive benefits and normative costs
	b.	normative benefits and positive costs
	c.	marginal benefit and marginal cost
	d.	opportunity cost and the sunk cost

1 points

QUESTION 13

If x increases whenever y decreases, then x and y are

	a.	positively related.
	b.	negatively related.
	c.	not related.
	d.	related but whether positively or negatively related depends on whether the x variable or the y variable is plotted on the vertical axis.

1 points

QUESTION 14

Rational choice

	a.	is what you must give up get what you want.
	b.	is made by comparing different incentives.
	c.	making the choice.
	d.	provides the answer to only the "how" question.

1 points

QUESTION 15

Scarcity results from the fact that

	a.	choices made in self-interest rare not always in the social interest.
	b.	not all goals are desirable.
	c.	people's wants exceed the resources available to satisfy them
	d.	we cannot answer the major economic questions.

1 points

QUESTION 16

When Delta decides to quit flying to Lithuania, it directly answers the ________ question.

	a.	what
	b.	why
	c.	for whom
	d.	how

1 points

QUESTION 17

A normative statement

	a.	says what is currently believed about the way the world operates.
	b.	depends on someone's values.
	c.	cannot use the word "should."
	d.	must be tested to determine if it is correct.

1 points

QUESTION 18

Which of the following statements best describes the study of economics? Economics studies how

	a.	to organize production so that scarcity does not occur.
	b.	to create incentives so that scarcity does not exist.
	c.	firms make profits.
	d.	we make choices in the face of scarcity.

1 points

QUESTION 19

Which of the following best defines economics?

	a.	Economics teaches how to limit our wants.
	b.	Economics is concerned with prices and quantities of goods and services, both at the individual level and at the industry level.
	c.	Economics helps you earn as much money as possible.
	d.	Economics studies how to choose the best alternative when coping with scarcity.

1 points

QUESTION 20

In economics, cost is ________, and benefit is ________.

	a.	the amount of money that you pay for something; the amount of money that someone else is willing to pay you
	b.	what you are willing to pay on the margin; what the government pays you when you are unemployed or retired
	c.	what you must give up to get something; what you are willing to give up to get it
	d.	the amount of money that you pay on the margin; the amount of money that you receive on the margin

1 points

QUESTION 21

If Jessie studies economics for two hours instead of going to the movies with her friends, then

	a.	the opportunity cost of studying is the missed movie.
	b.	Jessie is ignoring a sunk cost
	c.	Jessie is not responding to any incentives.
	d.	Jesse definitely is making a rational choice.

1 points

QUESTION 22

Going skiing will cost Adam $80 a day. He also loses $40 per day in wages because he has to take time off from work. Adam still decides to go skiing.

	a.	The $80 price of skiing is a sunk cost and so did not affect Adam's decision.
	b.	He loses a total of $120 per day, so his decision is irrational.
	c.	His decision is rational if Adam's marginal benefit of spending a day skiing is greater than his marginal cost.
	d.	Adam's lost $40 per day in wages is a sunk cost and so did not affect his decision.

1 points

QUESTION 23

If a curve falls and then rises, it

	a.	has a maximum.
	b.	has a linear relationship.
	c.	has a minimum.
	d.	has a constant slope relationship.

1 points

QUESTION 24

In 2003, Michael Jordan decided to retire from professional basketball. Which of the following was an opportunity cost of his decision?

	a.	the avoidance of pain and injury from playing one more year
	b.	the time he could spend with his family while retired
	c.	the lost salary from playing one more year
	d.	the value of new employment opportunities that would await him

1 points

QUESTION 25

Which of the following statements is a normative statement?

	a.	Military spending as a percent of government spending decreased by 5 percent in the 1990s.
	b.	Every American household should have health care insurance coverage.
	c.	Welfare reform has decreased the amount the government spends on welfare.
	d.	The price of computers fell last year.

greencat417

1.Write a function encipher(s, n) that takes as inputs an arbitrary string s and a non-negative integer n between 0 and 25, and that returns a new string in which the letters in s have been "rotated" by n characters forward in the alphabet, wrapping around as needed. For example:

>>> encipher('hello', 1) result: 'ifmmp'

>>> encipher('hello', 2) result: 'jgnnq'

>>> encipher('hello', 4) result: 'lipps'

Upper-case letters should be "rotated" to upper-case letters, even if you need to wrap around. For example:

>>> encipher('XYZ', 3)

result: 'ABC'

Lower-case letters should be "rotated" to lower-case letters:

>>> encipher('xyz', 3)

result: 'abc'

Non-alphabetic characters should be left unchanged:

>>> encipher('#caesar!', 2)

result: '#ecguct!'

Hints/reminders:

You can use the built-in functions ord and chr convert from single-character strings to integers and back:

>>> ord('a')

result: 97

>>> chr(97)

result: 'a'

You can use the following test to determine if a character is between 'a' and 'z' in the alphabet:

if 'a' <= c <= 'z':

A similar test will work for upper-case letters.

We recommend writing a helper function rot(c, n) that rotates a single character c forward by n spots in the alphabet. We have given you a template for this helper function in ps3pr3.py that checks to ensure that c is a single-character string. We wrote rot13(c) in lecture; rot(c, n) will be very close to rot13(c)! You can test your rot(c, n) as follows:

>>> rot('a', 1)

result: 'b'

>>> rot('y', 2)

result: 'a'

>>> rot('A', 3)

result: 'D'

>>> rot('Y', 3)

result: 'B'

>>> rot('!', 4)

result: '!'

Once you have rot(c, n), you can write a recursive encipher function.

Once you think you have everything working, here are three more examples to try:

>>> encipher('xyza', 1)

result: 'yzab'

>>> encipher('Z A', 2)

result: 'B C'

>>> encipher('Caesar cipher? I prefer Caesar salad.', 25)

result: 'Bzdrzq bhogdq? H oqdedq Bzdrzq rzkzc.'

2.Write a function decipher(s) that takes as input an arbitrary string s that has already been enciphered by having its characters "rotated" by some amount (possibly 0). decipher should return, to the best of its ability, the original English string, which will be some rotation (possibly 0) of the input string s. For example:

>>> decipher('Bzdrzq bhogdq? H oqdedq Bzdrzq rzkzc.')

result: 'Caesar cipher? I prefer Caesar salad.'

Here are two more examples:

>>> decipher('Hu lkbjhapvu pz doha ylthpuz hmaly dl mvynla lclyfaopun dl ohcl slhyulk.')

result: 'An education is what remains after we forget everything we have learned.'

>>> decipher('python')

result: 'eniwdc'

------------------------------------

def rot(c, n):
""" your docstring goes here """
# check to ensure that c is a single character
assert(type(c) == str and len(c) == 1)

def letter_prob(c):
""" if c is the space character (' ') or an alphabetic character,
returns c's monogram probability (for English);
returns 1.0 for any other character.
adapted from:

"""
# check to ensure that c is a single character
assert(type(c) == str and len(c) == 1)

if c == ' ': return 0.1904
if c == 'e' or c == 'E': return 0.1017
if c == 't' or c == 'T': return 0.0737
if c == 'a' or c == 'A': return 0.0661
if c == 'o' or c == 'O': return 0.0610
if c == 'i' or c == 'I': return 0.0562
if c == 'n' or c == 'N': return 0.0557
if c == 'h' or c == 'H': return 0.0542
if c == 's' or c == 'S': return 0.0508
if c == 'r' or c == 'R': return 0.0458
if c == 'd' or c == 'D': return 0.0369
if c == 'l' or c == 'L': return 0.0325
if c == 'u' or c == 'U': return 0.0228
if c == 'm' or c == 'M': return 0.0205
if c == 'c' or c == 'C': return 0.0192
if c == 'w' or c == 'W': return 0.0190
if c == 'f' or c == 'F': return 0.0175
if c == 'y' or c == 'Y': return 0.0165
if c == 'g' or c == 'G': return 0.0161
if c == 'p' or c == 'P': return 0.0131
if c == 'b' or c == 'B': return 0.0115
if c == 'v' or c == 'V': return 0.0088
if c == 'k' or c == 'K': return 0.0066
if c == 'x' or c == 'X': return 0.0014
if c == 'j' or c == 'J': return 0.0008
if c == 'q' or c == 'Q': return 0.0008
if c == 'z' or c == 'Z': return 0.0005
return 1.0

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MGEB11H3 Lecture Notes - Lecture 8: Random Variable, Sample Space, All The Same

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