Discrete Random Variables

A random variable is a quantity whose future outcomes are uncertain.

A discrete random variable can take on at most a countable number of possible values.

Cannot count the outcomes of a continuous random variable. (eg. rate of return)

The probability function specifies the probability that the random variable takes on a specific value: P(X = x).

For a discrete random variable, p(x) = P(X= x).

For continuous random variables, the probability function is denoted f(x) and called the probability density function (pdf).

Discrete uniform distribution has a finite number of specified outcomes, and each outcome is equally likely.

A probability function has two key properties:

(1) 0 ≤ p(x) ≤ 1;

(2) The sum of the probabilities p(x) over all values of X equals 1.

The cumulative distribution function (cfd), or distribution function, gives the probability that a random variable X is less than or equal to a particular value x, P(X ≤ x).

For both discrete and continuous random variables, the shorthand notation is F(x) = P(X ≤ x).

The cdf has two other characteristic properties:

The cdf lies between 0 and 1 for any x: 0 ≤ F(x) ≤ 1.

As we increase x, the cdf either increases or remains constant.