Continuous Random Variables and Probability Distributions

Probability Density Function:

Let X be a continuous random variable, and let x be any number lying in the range of values this random variable can take. The probability density function, f (x), of the random variable is a function with the following properties:

1.    f(x) > 0 for all values of x.
2.    The area under the probability density function, f(x), over all values of the random variable, X, is equal to 1.0.
3.    Suppose that this density function is graphed. Let a and b be two possible values of random variable X, with a < b. Then the probability that X lies between a and b is the area under the density function between these points.
4.    The cumulative distribution function, F(), is the area under the probability density function , f(x), up to  :

                         
where Xm  is the minimum value of the random variable x.

Areas Under Continuous Probability Density Functions

Let X be a continuous random variable with probability density function f(x) and cumulative distribution function F(x). Then the following properties hold:

1.    The total area under the curve f(x) is 1.
2.    The area under the curve f(x) to the left of  is F(), where  is any value that the random variable can take.