Expected value Law of Averages Central Limit Theorem

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Expected value Law of Averages Central Limit Theorem - PowerPoint PPT Presentation

Chapters 16, 17, and 18. Expected value Law of Averages Central Limit Theorem. Box Models. Flipping a coin n times, or rolling the same die n times, or spinning a roulette wheel n times, or drawing a card from a standard deck n times with replacement , …

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Chapters 16, 17, and 18

Expected valueLaw of Averages Central Limit Theorem

BoxModels
• Flipping a coin n times, orrolling the same die n times, or

spinning a roulette wheel n times, or drawing a card from a standard deck n times with replacement, …

• Interested in the accumulation of a certain quantity? We can box model the process.
• Actual outcomes are therefore abstractly represented by tickets.
How to Make a Box Model
• First draw a rectangular box.
• Then write next to the box how many times you are drawing from it: n = …
• What tickets go inside the box?
• That depends on what valueyou could add on to a requested quantity each time you draw from the box!
• Then, write the probability of drawing a particular ticket next to that ticket.
• Examples: Chapter 16, #5-8
ExpectedValue
• The expected value of n draws from the box is therefore given by:
• EVn = n*EV1
• The expected value of 1 draw from the box, also called the box average, is given by:
• EV1= weighted average of tickets in box
• = first ticket *probability of drawing first ticket + second ticket *probability of drawing second ticket + …
Law of Averages
• “The more you play a box-model-appropriate game, the more likely you get what you see of the box.”
• “What is expected to happen will happen.”
• Examples: Chapter 16, #1, #4
• A consequence of the Law of Averages is that we should not hope to come away with a gain by playing many times – we will eventually come out as a loser if we play long enough.
Standard error
• The expected value of n draws is given by EVn = n*EV1, where EV1 is the average of the box.
• Now of course our actual accumulated total could differ somewhat from the expectation, and we call our typical deviation standard error, given by:
• SEn = √n *SE1, where SE1 is the standard error of the box.
Standard error of a box
• Standard error of a box, or standard error of a single play, or standard error of a single draw, all mean the same thing.
• For a box with only two kinds of tickets, valued at A and B respectively, and with probability of p and q of being drawn respectively, the standard error of the box is given by:
• SE1=|A-B|* √(p*q)
• Examples: Chapter 17 #10
ContinuityCorrection
• This is related to the normal table we played with.
• Now the EVn acts as the “Average”
• And SEn acts as the “Standard Deviation”
• Chapter 17, Question 3c
• Continuity Correction is needed when you are dealing with discrete outcomes.
• Suggestion: Draw the normal curve and label the average. Then judge where you want to be and in what direction you should shade; then standardize and look up percentages.
• And so the new version of Standardization Formula:
C.C. and Discrete Outcomes
• When do we know we may use continuity correction?
• That’s when the observed outcomes are discrete.
• For example, if you are counting the number of democrats among a sample of 400 people, you can probably get 0, 1, 2, …, 399 ,or 400, but nothing else between any two numbers (such as 349.97)
• Examples: All questions in Chapter 18 where the box model is a “COUNTING BOX” (box with only 0 and 1)
• A non-example: Height of people in the US
• Example: P(at least break even) = P(actual > -0.5)
• Example: P(lose more than \$10) = P(actual < -10.5)
• Example: P(win more than \$20) = P(actual >20.5)
• Example: P(no more than 2300 heads) = P(actual < 2300.5)
Central limit theorem (CLT)
• The basis for what we did is called

Central Limit Theorem.

• The Central Limit Theorem (CLT) states that if…
• We play a game repeatedly
• The individual plays are independent
• The probability of winning is the same for each play
• Then if we play enough, the distribution for the total number of times we win is approximately normal
• Curve is centered on EVn