ECO220Y1 Lecture Notes - Lecture 11: Probability Distribution, Standard Deviation, Graduate Management Admission Test

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13 Jun 2018

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ECO220

Lecture 11

June 12, 2018

Chapter 9 (Cont.): Probability Distributions

Let Z ~ N(0, 1)

We covered:

(a) P(Z  0.8) = ?

(b) P(0  Z  k) = 0.3, k = ?

Theorem: Let X ~ N(, = variance)

Then 

 ~ N(0, 1)

Example 1: Let X ~ N(50,  = 10)

(a) P(X  60) = P( 

  

 ) = P(Z  1) = 0.1587

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ECO220Y1 Full Course Notes

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Example 1: let x ~ n(50, = 10) (a) p(x (cid:3410) 60) = p( (cid:3410) (cid:2874)(cid:2868) (cid:2873)(cid:2868)(cid:2869)(cid:2868) ) = p(z (cid:3410) 1) = 0. 1587. Example 2: suppose gmat has an average score of 600 and standard deviation 100. June 12, 2018 (c) three students are writing the gmat. What is the probability that at least 2 of them have. = p(z > 1) = 0. 5 - 0. 3413 = 0. 1587 (cid:4667) We will approximate the numerical answer using the normal table. Normal approximation to binomial (d) suppose 500 students are writing the gmat. What is the probability that at least 300 of them have gmat scores over 700? p = p(x > 700) = 0. 1587. P(x (cid:3410) 300) = p(x = 300, 301, 302, 500) Let x ~ b(n = 500, p = 0. 4: the mean is at 200, the longest line will be at 200. In this example, find p(x > 300) = p(x = 301, 302, 500)

Related Questions

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

olivemink636

(Please show as much work as possible)

1. You are bidding in a second-price auction for a painting that you value at $800. You estimate that other bidders are most likely to value the painting at between $200 and $600. Which of these is likely to be your best bid?

a. $1,000

b. $800

c. $600

d. $400

2. Which of the following is true about different ways of conducting a private-value auction?

a. A first-price auction is strategically equivalent to a second-price auction.

b. A first-price auction is strategically equivalent to an English auction.

c. A second-price auction is strategically equivalent to an English auction.

d. None of the above

3. Suppose that five bidders with values of $500, $400, $300, $200, and $100 attend an oral auction. Which of these is closest to the winning price?

a. $500

b. $400

c. $300

d. $200

4. In the above auction, if the bidders with the first- and third-highest values ($500 and

$300) collude, which of these is closest to the winning price?

a. $500

b. $400

c. $300

d. $200

5. If a seller is concerned about collusion among bidders, which of the following changes to the auction, should the seller make?

a. Hold frequent, small auctions instead of infrequent large auctions.

b. Conceal the amount of winning bids.

c. Publically announce the name of each auction's winner.

d. Hold a second-price instead of a first-price auction.

6. You're holding an auction to license a new technology that your company has developed. One of your assistants raises a concern that bidders' fear of the winner's curse may encourage them to shade their bids. How might you address this concern?

a. Release your analyst's positive scenario for the technology's future profitability.

b. Release your analyst's negative scenario for the technology's future profitability.

c. Use an oral auction.

d. All of the above

7. In a first-price auction, you bid ________ your value, and in a second-price auction you bid _________ your value.

a. at; above

b. below; above

c. below; at

d. below; below

8. You hold an auction among three bidders. You estimate that each bidder has a value of either $16 or $20 for the item, and you attach probabilities to each value of 50%. What is the expected price? If two of the three bidders collude, what is the price?

9. In Sweden, firms that fail to meet their debt obligations are immediately auctioned off to the highest bidder. (There is no reorganization through Chapter 11 bankruptcy.) The current managers are often high bidders for the company. Why?

10. When a famous painting becomes available for sale, it is often known which museum or collector will be the likely winner. Yet, representatives of other museums that have no chance of winning are actively wooed by the auctioneer to attend anyway. Why?

11. The deities Mars and Venus often do battle to create the weather conditions on Earth. Venus prefers extreme temperatures (especially heat), while Mars prefers temperate conditions. The payoffs (expressed in Points of Wrath) are given below.

		Venus
		Warm	Chill
Mars	Warm	20 , 0	0 , 10
Chill	0 , 90	20 , 0

What is the unique mixed-strategy equilibrium of the above game?

(Let p be the probability of "Warm" for Mars, and q the probability of "Warm" for Venus.)

a) p=9/10, q=1/2

b) p=1/2, q=1/10

c) p=1/2, q=1/2

d) p=1/10, q=1/10

Player 2

		H	D
Player 1	H	0 , 0	4 , 1
D	1 , 4	2 , 2

12. The above game is the title of the hawk-dove game and used by evolutionary biologists to describe evolutionary processes. It is also used to model how a business should grow. In the above game, what is the Nash equilibrium in pure strategies and mixed strategies.?

Assume the cost of producing the goods is zero and that each consumer will purchase each good as long as the price is less than or equal to value. Consumer values are the entries in the table.

	Good 1	Good 2
Consumer A	$2,300	$1,700
Consumer B	$2,800	$1,200

13. Suppose the monopolist only sold the goods separately. What price will the monopolist charge for good 1 to maximize revenues for good 1?

a. $2,300

b. $2,800

c. $1,200

d. $1,700

14. What is the total profit to the monopolist from selling the goods separately?

a. $4,500

b. $6,300

c. $7,000

d. $6,000

15. What is a better pricing strategy for the monopolist? At this price, what are the total profits to the monopolist?

a. Bundle the goods at $2,800; Profits = $5,600

b. Bundle the goods at $4,000; Profits = $8,000

c. Charge $2,800 for good 1 and charge $1,700 for good 2; Profits = $4,500

d. Charging the lowest price for each good individually is the best pricing strategy; Profits = $7,000

16. The prisoners' dilemma is an example of

a. a sequential game.

b. a simultaneous game.

c.a shirking game.

d. a dating game

17. Nash equilibrium

a. is where one player maximizes his payoff, and the other doesn't.

b. is where each player maximizes his own payoff given the action of the other player.

c.is where both players are maximizing their total payoff.

d. is a unique prediction of the likely outcome of a game.

viridianlocust640

Match the definitions or complete the statements in the first part of the quiz with the appropriate terms found in the second part.

Definitions/Statements

1. _____ refers to marketersâ focus on a product rather than on the needs of the market.

2. _____ companies do not try to persuade consumers to buy what the firm has already produced, but rather to produce only products that they know they can sell, thereby satisfying consumersâ needs and turning them into loyal customers.

3. The impact of technology on buyer behavior relates to factors such as ____; the use of technology to engage customers and build relationships; customization of products and promotional messages; and better prices and distribution.

4. In the case of situational influences, this relates to the situation in which or the occasion on which the product will be used, e.g., special occasion or everyday use.

5. In the case of situational influences, this relates to the fact if we are in a store, point-of-purchase displays could influence us to buy something. We could also make unplanned purchases because we remembered something from seeing the brand while we are in the store.

6. This discipline on which consumer researchers draw focuses on society and large groups in society such as family and social class.

7. This discipline on which consumer researchers draw focuses on cultures and differences among cultures.

8. These are temporary circumstances that affect consumer purchase decisions.

9. In the case of situational influences, this refers to the fact that the person and the occasion for which we are buying something influence our purchase decisions.

10. According to the simple model of consumer decision making, reference groups, family, social class, culture, and subculture are among the _____ on buyer behavior.

11. According to the simple model of consumer decision making, motivation, personality trait, perception, and attitudes are among the _____ on buyer behavior.

12. According to the simple model of consumer decision making, advertising, buzz agents, customized messages, social media, and word-of-mouth are examples of ____.

13. This discipline on which consumer researchers draw focuses on behavior in small groups and helps consumer researchers to understand issues such as opinion leadership, diffusion of innovations, and communication.

14. In the case of situational influences, ______ are consumer mood states or physiological state (for example, hunger, thirst) that influence consumer decision-making.

15. The selection of one or more segments by a marketer to serve.

16. Developing a distinct image for a brand in the mind of consumers.

17. This is an effort by a marketer to change the existing perception that consumers have of a brand.

19. ___ segmentation involves segmenting based on variables such as urban, rural, suburb, state, cities, counties, and regions.

20. ____ segmentation is based on variables such as consumersâ lifestyles as indicated by their activities, interests, and opinions.

21. This criterion for effective targeting says that marketers should have some way of deciding which consumers fall into which segments.

22. This criterion for effective targeting says that marketers should be able to reach segments with appropriate media and distribution channels.

23. This is the most popular segmentation system combining lifestyles and values of consumers.

24. This household segmentation model is based on a combination of geographic and demographic information and the principle that âBirds of a feather flock together.â

25. ______ is defined as âsending consumers personalized and prompt offers and promotional messages designed to reach the right consumers and deliver to them highly relevant messages at the right time and more accurately than when using conventional segmentation techniques.â

Terms

A. Positioning

B. Accessibility

C. Anthropology

D. Situational influences

E. Consumption situation

F. Task definition

G. Identification

H. Marketing-oriented

I. Targeting

J. Psychographic

K. Marketing myopia

L. PRIZM

M. Interactive and novel communication channels

N. Sociology

O. Demographic

P. Purchase situation

Q. Behavioral targeting

R. Antecedent states

S. Communication sources

T. Geographic

U. Sociocultural influences

V. VALS

W. Psychological influences

X. Social psychology

Y. Repositioning

ECO220Y1 Lecture Notes - Lecture 11: Probability Distribution, Standard Deviation, Graduate Management Admission Test

ECO220Y1 Full Course Notes

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