Continuous Probability Distributions

An example:
Suppose the random variable X represents the amount of rainfall in Bainbridge GA. in inches, for the year 2003. Further  suppose  that   X can take on any value between 1.0 inches and 57.8 inches (1.0 in.< X < 57.8 in). Lets calculate the probability that X will be exactly 25.4 inches in 2003.
P( X=25.8 in ) = ?

Thinking about the formula for calculating probabilities which tells us to divide the number of ways X can be 25.8 inches ( exactly one way) by the total number of outcomes in our sample space of numbers from 1.0 to 57.8 (an infinity of numbers), we would divide 1 by infinity.

 P (X=25.8) = 1/infinity = 0

The lesson to be learned is that if X is a continuous random variable then the probability that X takes on a specific value  is zero and it does us no good to speak of  probabilities at specific values for continuous random variables. For continuous random variables we speak of probabilities over intervals of numbers. In the above example we might ask for the probability that the number of inches of rainfall at Bainbridge in 2003 is between 10.2 and 17.5 inches. (e.g. P( 10.2< X < 17.5)

You may be wondering, how are we to solve this problem which essentially involves dividing infinity by infinity. Don't despair. This problem has been worked out for us by employing the worlds greatest calculating system, the calculus. No, you do not have to understand the calculus. All this has been worked out for us and presented  in such a way that anyone who wishes to work with continuous random variables may do so. . The mechanism for accomplishing this task is a  function called a density function. Associated with every continuous random variable is a density function . Finding probabilities of continuous random variables over an interval involves finding areas under a density function, and, finding areas under a density function means that we look these numbers in a table associated with the density function. 

Suppose the density function for rainfall at Bainbridge GA. in 2003 was given by the smooth curve shown below where x1 = 10.2 inches and x2 =17.5 inches. The probability that X lies between 10.1 inches and 17.5 inches is given by the area of the shaded region under the density function.

P( 19.1 <  X < 17.5) = area of shaded region

Now, to find the number associated with this area we would do one of two things.

   (1) Use the tools of the calculus   Or  (2) Look up the area in a table

Since you are not students of the calculus, we will of course use tables.

Since probabilities are numbers between zero and one, this means that the total area under a density function from end to end must be one. You will need to remember this!

Now lets turn our attention to the most famous family of continuous distribution , The family of  Normal Curves.

The Family of Normal Curves

Similarities of the family of normal curves:

Differences in the family of normal curves

Changing either or both the mean mu or the standard deviation sigma will result in a different member of the family.

The example below shows how changes in the mean mu without changes in the standard deviation sigma affect the graph.

Next we see how changes in the standard deviation sigma without changes in the mean Mu affect the graph.

We say that the family of normal curves are a two parameter family. This means their shape depends upon two parameters, the mean mu and the standard deviation sigma.

To see an interactive demonstration of this idea, click on the link below.

The Empirical Rule

Approximately 68% of  all observations fall between one SD from the mean.
Approximately  95 % of all observations fall between two SD from the mean.
Approximately 99.7% of all observations fall between three SD from the mean.