3.1 Random Variables (RVs)

What is a Random Variable (RV)?
A RV is any rule that associates a number with each sample point in a given sample space of some experiment.

What is the domain of a Random Variable (RV)?
The domain of an RV is the sample space.

What is the range of a Random Variable (RV)?
The range of an RV is the set of real numbers.

What is a realization or observed value of a Random Variable (RV)?
A realization or observed value of an RV is a particular value that an RV may return.

What is a discrete random variable?
A discrete random variable is a random variable whose possible realizations are either a finite set or a countably infinite set.

What is a continuous random variable?
A continuous random variable is a random variable whose possible realizations are an uncountably infinite set.

How do you denote a random variable?
You denote a random variable with an uppercase letter.

How do you denote a realization of a random variable?
You denote a realization of a random variable with the lowercase version of the random variable.

...

3.2 Probability Distributions for Discrete Random Variables

What is the probability that a discrete random variable takes on the value ?
The probability that a discrete random variable takes on the value is the sum of probabilities corresponding to all sample points () in that are assigned the value .

What is the definition of the Probability Mass Function (PMF) for ?
The definition of the PMF for is:

What does the Probability Mass Function (PMF) for do?
The PMF for assigns probabilities to each value of the random variable .

What is a Probability Mass Function (PMF)?
A PMF is a listing of possible values of a discrete random variable along with the probability of each value.

How can the probability distribution for a discrete random variable be represented?
The probability distribution for a discrete random variable can be represented by a formula, table, or graph that provides for all values.

What two things must be true for any discrete probability distribution?
The two things which must be true for any discrete probability distribution are:

  1. for all .
  2. where the summation is over all values of with nonzero probability.

...