Random Variables - Continuous
A Random Variable is a set of possible values from a random experiment.
Example: Tossing a coin
We could get Heads or Tails.
Let's give them the values Heads=0 and Tails=1 and we have a Random Variable X:
In short:
X = {0, 1}
Note: We could choose Heads=100 and Tails=150 or other values if we want! It is our choice.
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 continuous distribution.
Between a and b, all values of the random variable are equally likely.
The probability density between a and b is a constant value p
Total probability must be 1, so the area of the rectangle must be 1:
We can write:
f(x) = 1b−a for a ≤ x ≤ b
f(x) = 0 otherwise
Example: Old Faithful erupts every 91 minutes. You arrive there at random and wait for 20 minutes ... what's the probability you will see it erupt?
This is actually easy to calculate, 20 minutes out of 91 minutes is:
p = 20/91 = 0.22 (to 2 decimals)
But let's use the Uniform Distribution for practice.
To find the probability between a and a+20, find the blue area:
So there's about a 0.22 probability (20% chance) you will see Old Faithful erupt.
If you waited the full 91 minutes you would be sure (p=1) to have seen it erupt.
But remember this is a random thing! It might erupt the moment you arrive, or any time in the 91 minutes.
Cumulative Uniform Distribution
The Uniform Distribution can also be written as a cumulative (adding up as it goes along) distribution:
The probability starts at 0 and builds up to 1
This is called the Cumulative Distribution Function, or CDF.
F(x) is the standard notation for CDF.
Example (continued):
Let's use the CDF of the previous Uniform Distribution to work out the probability:
At a+20 the probability has accumulated to about 0.22
Other Distributions
| Knowing how to use the Uniform Distribution helps when dealing with more complicated distributions like this one: |  |
The general name for any of these is probability density function or pdf
The Normal Distribution
The most important continuous distribution is the Standard Normal Distribution
It has mean 0 and standard deviation 1, and its random variable is given the special letter Z.
The graph for Z is a symmetrical bell-shaped curve:
We usually want to find the probability that Z lies between two values.
Example: P(0 < Z < 0.45)
(What's the probability that Z is between 0 and 0.45)
This is found by using the Standard Normal Distribution Table
Go to row 0.4, then across to 0.05 (0.4+0.05=0.45) to find 0.1736:
P(0 < Z < 0.45) = 0.1736
Summary
- A Random Variable assigns numbers to outcomes of a random experiment
- Random Variables can be discrete or continuous
- A good example of a continuous Random variable is the Standard Normal variable Z