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Expected value Law of Averages Central Limit TheoremPowerPoint Presentation

Expected value Law of Averages Central Limit Theorem

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### 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 + …

- EV1= weighted average of tickets in box

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
- Spread measure is SEn

- Also holds if we are counting money won
- Note: CLT only applies to sums! See Chapter 18 Question 10.

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