Section 8.5 - Mean and Standard Deviation of a Probability Model

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Section 8.5 - Mean and Standard Deviation of a
Probability Model

• Special Topics

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Mean of a Probability Model

• The mean of a set of observations is the
  ordinary average.
• The mean of a probability model is also an
  average, but with an essential change, not all
  outcomes are equally likely.
• It is actually a weighted average.

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Formula for Mean of a Discrete Probability Model

• If the probability distribution of a probability
  model is as follows
• To find the mean (AKA expected value), multiply
  each possible value by its probability, then add
  all the products.

Value of X x1 x2 x3 xn
Probability p1 p2 p3 pn
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Calculator Shortcut
EX The distribution of the count of heads in 4
tosses was found to be
0(.0625) 1(.25) 2(.375) 3(.25) 4(.0625)
2 Put X in L1 and P(X) in L2, in L3 (L1 x L2).
You then can sum this total to get the mean or
expected value. You can also run a 1VARS Stats
on L1,L2 and it will produce the expected value.
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Standard Deviation of a Discrete Probability Model

• If the probability distribution of a probability
  model is as follows
• To find the standard deviation of the model

Value of X x1 x2 x3 xn
Probability p1 p2 p3 pn
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Calculator

• L1 Put the values of the random variable
• L2 Put the probabilities of each value
• At this point you can run a 1VARS Stats L1, L2
• it will give you the expected value and the
  standard deviation.

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Mean of a Continuous Probability Model

• What about continuous probability models? Think
  of the area under a density curve as being cut
  out of solid homogenous material. The mean µ is
  the point at which the shape would balance. This
  is what this idea looks like with a skewed model

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Mean of a Continuous Probability Model

• When the model is symmetric (normal, uniform, or
  other symmetric shape), the mean (and the median)
  lies at the center of the curve.
• Mean
• Median

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The Law of Large Numbers
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The Law of Large Numbers

• The law of large numbers tells us that in many
  trials the proportion of trials on which an
  outcome occurs will always approach its
  probability.
• The law of large numbers also explains why
  gambling can be a business. The winnings (or
  losses) of a gambler on a few plays are
  uncertainthat's why gambling is exciting. It is
  only in the long run that the mean outcome is
  predictable. The house plays many tens of
  thousands of times. So the house, unlike
  individual gamblers, can count on the long-run
  regularity described by the law of large numbers.

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Homework

• Worksheet 8.5