Psychology 202aAdvanced 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