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