SUBMATH: RANDOM AND CONTINUOUS VARIABLES

Random Variables – Continuous

A Random Variable is a set of possible values from a random experiment.

Continuous

Random Variables can be either Discrete or Continuous:

• Discrete Data can only take certain values (such as 1,2,3,4,5)
• Continuous Data can take any value within a range (such as a person’s height)

In our Introduction to Random Variables (please read that first!) we look at many examples of Discrete Random Variables.

But here we look at the more advanced topic of Continuous Random Variables.

The Uniform Distribution

The Uniform Distribution (also called the Rectangular Distribution) is the simplest distribution.

It has equal probability for all values of the Random variable between a and b:

The probability of any value between a and b is p

We also know that p = 1/(b-a), because the total of all probabilities must be 1, so

We can write:

P(X = x) = 1/(b−a) for a ≤ x ≤ b
P(X = x) = 0 otherwise

Cumulative Uniform Distribution

We can have the Uniform Distribution as a cumulative (adding up as it goes along) distribution:

The probability starts at 0 and builds up to 1

Summary

• A Random Variable is a variable whose possible values are numerical outcomes of a random experiment.
• Random Variables can be discrete or continuous.
• An important example of a continuous Random variable is the Standard Normal variable, Z.

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Discrete Random Variables

Discrete random variables can take on either a finite or at most a countably infinite set of discrete values (for example, the integers).

Their probability distribution is given by a probability mass function which directly maps each value of the random variable to a probability.

For example, the value of ${\text{x}}_1$ takes on the probability ${\text{p}}_1$, the value of ${\text{x}}_2$ takes on the probability ${\text{p}}_2$, and so on.

The probabilities ${\text{p}}_{\text{i}}$ must satisfy two requirements: every probability ${\text{p}}_{\text{i}}$ is a number between 0 and 1, and the sum of all the probabilities is 1. (${\text{p}}_1+{\text{p}}_2+\cdots +{\text{p}}_{\text{k}}=1$)

Examples of discrete random variables include the values obtained from rolling a die and the grades received on a test out of 100.

Continuous Random Variables