General Psychology Chapter 12 The Psychological Disorders Chapter 12 The Psychological Disorders.
Psychology 202a Advanced Psychological Statistics September 22, 2015.
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Transcript of Psychology 202a Advanced Psychological Statistics September 22, 2015.
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Psychology 202aAdvanced Psychological
Statistics
September 22, 2015
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The Plan for Today
• Wrapping up conditional distributions.
• Random variables and probability distributions.
• Continuous random variables.
• Rules for combining probabilities.
• (Bayes’ theorem.)
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New concept: random variables
• Review definition of variables and distributions
• New kind of variable:– imaginary– a set of values that could occur– random variable
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Distribution of a random variable
• Just as variables have distributions, so do random variables.
• The distribution of a random variable is the set of values that could occur if we were to observe the variable…
• …together with the long run relative frequencies with which those values would occur.
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Probability distributions
• This special type of imaginary distribution is called a probability distribution.
• Definition: A probability distribution is the set of values that could occur for a random variable, together with the long-run relative frequencies with which they do occur when that random variable is actually observed.
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Frequentist approach
• long-run relative frequency = probability
• The law of large numbers:– If a random process is observed repeatedly,
the proportion of times a particular outcome of that process occurs approaches the probability of the outcome as the number of repetitions becomes large.
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More detail on the frequentist approach
• Just as relative frequency observed in the long run is probability...
• ...descriptive statistics observed in the long run become parameters of the probability distribution,
• ...and graphics observed in the long run become pictures of the probability distribution.
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Some technical matters
• A well-defined outcome of a random variable is called an event.
• Random variables may be continuous or discrete.
• The probability of any particular outcome for a continuous random variable is zero.
• In such cases, events must be described in terms of ranges of possible values.
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Bernoulli trials: the simplest possible random variable
• On each repetition, one of two discrete values may occur, and the probability of each is the same on each trial.
• Examples:– tossing a coin– rolling a die– choosing a random person and observing that
person’s sex
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Bernoulli processes
• The name for that type of random variable is Bernoulli random variable or Bernoulli process.
• Think about the example of tossing a fair coin.
• digression on document camera and in R
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Recapping types of distributions• So far, we have discussed two types of
distributions• Distributions:
– Values that a variable takes on, with frequencies (or relative frequencies) of those values
• Probability distributions:– Values that a random variable could take on,
together with probabilities of those values
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Similarities
• We’ve seen that the same ways of thinking can help us understand the shape of both types of distribution.
• The trick to understanding probability distributions is to apply those ways of thinking to what would happen in the long run.
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Continuous random variables
• Examples:– Normal distribution– Uniform distribution
• New terminology: probability density function (abbreviated “pdf”)
• Investigating some properties of the uniform distribution