Psychology 202a
Advanced Psychological
Statistics
September 22, 2015
The Plan for Today
• Wrapping up conditional distributions.
• Random variables and probability
distributions.
• Continuous random variables.
• Rules for combining probabilities.
• (Bayes’ theorem.)
New concept: random variables
• Review definition of variables and
distributions
• New kind of variable:
– imaginary
– a set of values that could occur
– random variable
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.
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.
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.
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.
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.
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
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
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
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.
Continuous random variables
• Examples:
– Normal distribution
– Uniform distribution
• New terminology: probability density
function (abbreviated “pdf”)
• Investigating some properties of the
uniform distribution