1. Psychology 202a Advanced Psychological Statistics
September 14, 2017

2. The Plan for Today Scatterplots in R
Random variables and probability distributions.
Continuous random variables.
Rules for combining probabilities.
(Bayes’ theorem.)

3. New concept: random variables
Review definition of variables and distributions
New kind of variable:
imaginary
a set of values that could occur
random variable

4. 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.

5. 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.

6. 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.

7. 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.

8. 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.