Continuous-valued Random Variables

• So far we have focused on discrete(-valued) random variables, e.g. must be an integer.

• Examples of discrete random variables:
  • The number of arrivals in one second.

  • The number of attempts until success.

• A continuous-valued random variable takes on a range of real values, e.g. ranges from 0 to as s varies.

• Examples of continuous(-valued) random variables:
  • The time when a particular arrival occurs.

  • The time between consecutive arrivals.

• A discrete random variable has a ``staircase" CDF.

• A continuous random variable has (some) continuous slopes to its CDF.

• Thus, for a continuous random variable X, we can define its probability density function (pdf) by

  

• Note that since is non-decreasing in x, we have

  

• Properties of Continuous Random Variables
• Expectation of a Continuous Random Variable
• Important Continuous Random Variable: Exponential
• Memoryless Property of the Exponential