Rules for Means and Variances of Random Variables

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Rules for Means and Variances of Random Variables

• AP Statistics
• Chapter 7

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Rules for Means

• If X is a random variable and a and b are fixed
  numbers, then

• EXAMPLE
• Let X be a random variable representing outcome
  of a roll of a six-sided fair die
• What is the mean of X?
• We take the result of the roll, double it and add
  10.
• What is the mean of the new random variable?

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Rules for Means, continued

• If X and Y are random variables, then

• EXAMPLE
• You roll a die twice. What is the mean of the
  sum of the 2 dice?

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What about Variability?

• What happens to the standard deviation (and
  variance) when we add or subtract 2 random
  variables?
• THE SECOND MOST IMPORTANT CONCEPT OF THIS
  CLASS!!!!
• Drum Roll

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The Pythagorean Theorem of Statistics

• For independent random variables
• Var(X Y) Var(X) Var(Y)

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Pythagorean?

• In a right triangle,
• For independent random variables,

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Questions

• Why variances?
• Why must they be independent?
• Why add even for differences?
• Why does it matter?

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Why variances?

• Why not standard deviations?

Issue 1 How are we measuring variability?
Issue 2 Its that Pythagorean thing 3 4 7,
but
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Why independent?

• X During the last 24 hours, how many hours
  did you sleep?

• Y During the last 24 hours, how many hours
  were you awake?

Whats the standard deviation of X Y?
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Add even for the difference?

• Proof

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Seriously, why add?

• Case 1 Cereal
• Pour a 3-ounce bowl from a 16-ounce box.
• How much cereal is left in the box?
• How certain are you of that estimate?

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Is subtracting the same as adding?
Case 2 Fruit Grapefruit weigh 16-24 ounces each
and oranges weigh 9-14 ounces each. Pick one of
each fruit at random.

• Whats the possible range of the sum of the
  weights?

• Whats the possible range of the difference in
  weights?

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Why does it matter?

• Making Statistics make sense
• Understanding the Pythagorean Theorem of
  Statistics empowers students to deal effectively
  and insightfully with a myriad of issues that
  arise in Statistics.

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Why does it matter?

• Standard Deviation to Binomial Distribution
  (Chapter 8)
• Leads to the most important concept of this class
• Central Limit Theorem (Chapter 9)
• Inference (all of 2nd semester)
• Difference of means, two-sample tests, confidence
  intervals

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Rules for Variances

• Rule 1 If X is a random variable and a and b
  are fixed numbers

• EXAMPLE
• X is outcome of roll of die
• What is variance of X?
• We take the result of a roll, double it and then
  add 10.
• What is the variance of this result?

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Rules for Variances

• RULE 2 (The Pythagorean Theorem of Statistics)

• EXAMPLE
• We roll two dice.
• What is the standard deviation of the sum of the
  rolls?
• The difference of the rolls?