Class Notes, Class 3

This chapter formalizes ideas that we talked about last time:

Probability function (narrowed to random variable)

Discrete random variable / Probability (mass) function

Continuous random variable / Probability density function

Joint random variable, discrete or continuous

Independence for joint random variables, discrete or continuous

And one idea that we didn't talk about last time:

Cumulative probability function

Notice the two types of notation, p( ) and f( ). The p( ) notation always refers to probability per se. The f( ) notation refers to probability for discrete functions, but to probability density for continuous functions.

Also introduced is the F( ) notation, which always refers to cumulative probability.

Random variable

The elements of the sample space (outcomes) are real numbers.

The random variable is the outcome of the experiment

(def) A random variable is a function that associates a real number with each element in a sample space

Example: Bernoulli trial - an underlying distribution

Toss a fair coin; call "heads" 1 and "tails" 0

Outcomes, X: 0 or 1, therefore it is a random variable

Example: Binomial dstribution - a sampling distribution

Toss a fair coin 5 times

Random sample from a Bernoulli Process

(trials are independent, identically distributed, i.e. p is constant)

Random variable: count the number of heads

Outcomes, X : 0, 1, 2, 3, 4, 5

Discrete vs. continuous random variables

Discrete random variable

The sample space is finite or countably infinite.

Continuous random variable

The sample space includes an interval of the real numbers.

Discrete probability distributions

probability function, probability mass function, probability distribution, f(x)

Informally, f(x) = p(x)

Formal notation for random variable X:

Example: toss two coins, add up the number of heads

the probability mass function

Cumulative probability distribution - discrete random variable

cumulative distribution of a discrete random variable X, F(x)

Step function

Example: toss two coins, add up the number of heads

the cumulative probability function

Continuous probability distribution

Probability density function, density function, f(x)

f(x) is probability density; p(x) = 0

Example: uniform distribution from 0 to 1

the probability density function

Cumulative probability distribution

cumulative distribution of a continuous random variable X, F(x)

Example: uniform distribution from 0 to 1

the cumulative probability function

Example: p 90 and Figure 3.6: Example 3.12

Joint probability distributions (bivariate)

Discrete random variables

Joint probability distribution, joint probability mass function, f(x, y)

Marginal distributions, g(x) and h(y)

Conditional distributions, f(y|x) (i.e. f(y|X=x)) and f(x|y) (i.e. f(x|Y=y))

Independence

Example: p106 3.50 - also, are X and Y independent?

Continuous random variables

Joint density distribution, joint probability density function, f(x, y)

Marginal distributions, g(x) and h(y)

Conditional distributions, f(y|x) (i.e. f(y|X=x)) and f(x|y) (i.e. f(x|Y=y))

Independence

Multivariate independence

Example p106 3.55

Probability functions

discrete, continuous functions

definitions of discrete and continuous probability functions

cumulative probability function

joint probability function

marginal probability function

conditional probability function

independence