17.1Continuous random variables

For example, if X is the random variable which takes its values as ‘distance in metres’ that a parachutist lands from a particular marker, then X is a continuous random variable, and here the values which X may take are the non-negative real numbers.
The probability density function does not give probabilities and f (x) may take values greater than one. The probabilities are given by areas determined by the graph of f .
To determine the value of this probability you could begin by measuring the weight of a large number of randomly chosen people, and determine the proportion of the people in the group who have weights in this interval. Suppose after doing this a histogram of weights was obtained as shown.
In many cases the range of X is an unbounded interval, for example [1,∞) or indeed R. Therefore, some new notation is necessary.
Suppose that the random variable X has the density function with the rule:
C H A P T E R 17 Continuous random variables and their probability distributions
17.1 Continuous random variables
A continuous random variable is one that can take any value in an interval of the real number line.

Now, the probability of interest is no longer represented by the area under the histogram, but the area under the curve. That is:
If the range of the continuous random variable X is [a, b] then the domain of its probability density function f is [a, b]. The probability density function will satisfy two properties:
a Find the value of c that makes f a probability density function. b Find Pr(X > 1.5).
a Sketch the graph of f . b Show that f is a probability density function. c Find Pr(X > 0.5).