Probability distributions

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Probability distributions

• establishing link between probability theory and
  statistical judgments
• probability distribution graph showing the
  potential values of a variable and their
  corresponding probabilities

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• frequency histogram
• probability distribution

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Discrete variables

• only certain values (e.g. integers) can be
  values of the variable
• the sum of the heights of the bars is
• 1.0

• the expected value of a discrete variable
• E(X)

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Two theoretical discrete probability distributions

• 1 discrete uniform probability of each
  discrete outcome (of k outcomes) is equal.

a discreet uniform

• P(x) 1/k

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The Uniform Distribution

• A little simplistic and perhaps useless
• But actually well applied in two situations
• 1. The probability of each outcome is truly equal
  (e.g. the coin toss, card pick)
• 2. No prior knowledge of how a variable is
  distributed (i.e. complete uncertainty), the
  first distribution we should use is uniform (no
  assumptions about the distribution)

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P (head) 0.50 P (tail) 0.50
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Two theoretical discrete probability distributions

• 2 Discrete Rare Events --- Poisson
  distribution
• important in point pattern analysis
• forms the basis for models of randomly
  distributed points in an area
• tells the probability that a certain number of
  occurrences or points will fall in a certain unit
  of time or space

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Critical Information To Know

• Average Number of things in a unit
• Lamda or the Greek letter
• l

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Poisson distribution
e-l lx
where l is the average number of things in a unit
(time or area)
P(x)
x!
l 12/4 3
P(5) 2.71828335 / (54321)
0.1008 P(2) 2.71828332 / (21)
0.2240
you can look this up if you dont want to
calculate this! Table A2 in the back of the book
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What is the probability that we get 2 or 3 or 4
dots in a square?

• Probability of 4, plus the probability of 3, plus
  the probability of 2
• .168 .224 .224 .616

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What is the probability that we get 3 or 2 dots
in a square?

• Probability of 3 plus the probability of 2
• .224 .224 .448

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What is the least likely number of dots?

• Answer 12 (all dots in one square)
• Why?
• Probability of 12 is 0.001
• Note that this is smaller than the likelihood of
  no dots (0.0498)