Suppose we choose a real number at random between 0 and 1. Let *X*
be the number chosen. Then *X* is a continuous random variable with
the interval *(0,1)* as its range. Certain probabilities are obvious
here:

- 1.
- The probability that
*X*is between 0 and 1/2 is 1/2. - 2.
- The probability that
*X*is between 1/2 and 1 is 1/2. - 3.
- The probability that
*X*is between 0 and 1/4 is 1/4. - 4.
- The probability that
*X*is between 1/2 and 3/4 is 1/4. - 5.
- The probability that
*X*is between 1/8 and 2/8 is 1/8.

- 1.
- The probability that
*X*is between 1/4 and 3/4 is 1/2. - 2.
- The probability that
*X*is between 3/8 and 5/8 is 1/4. - 3.
- The probability that
*X*is between 7/16 and 9/16 is 1/8. - 4.
- The probability that
*X*is between 15/32 and 17/32 is 1/16.

For a continuous random variable *X* whose range is the interval
*(c,d)* the **probability model** of *X* is a curve *f(x)* such that
the probability that *X* is between *a* and *b* is the area
under the curve between *a* and *b*, that is, for some function *f(x)* the
*P*(*a* < *X* < *b*) is the area under the curve as shown in Figure 5.1

Notice that *f(x)* cannot be negative and that the total area under
the curve must be one.

Consider the above example where *X *is a number chosen at random
between 0 and 1. If we draw a straight line with slope 0 and height 1 above
the interval *(0,1)*, then the area under this line over the interval
*(a,b)*
is
,
which is our desired probability. The curve is given
in Figure 5.2. This is called the **uniform probability model** .

Here's a second example. Suppose we choose a point at random inside the unit circle, a circle with radius 1 and center at the origin, (sketch it!).

Let *X *be the distance between the chosen point and the origin, (sketch it!). See my sketch in Figure 5.3. Then the range of *X* is between 0 and 1, just like the uniform. But the probabilities are unlike the uniform. For example, it
is much more likely that *X *is between 3/4 and 1 than between 0 and
1/4, (Why? Sketch it!). In fact, you can show that the probability model
for *X *is a line over the interval *(0,1)* with intercept at
the origin and slope 2 (a triangle!, sketch it!). The graph is

Recall that the area of a triangle is
base
height. Show that the area of the triangle is 1. Next shade in the area under the curve from 1/4 to 3/4. Determine this area. You have just found
*P*(1/4 < *X* < 3/4).

- 1.
- Let X be a number chosen at random between 8 and 10. Determine the probabilities that X is between 8.5 and 9.5 and X is between 8.5 and 8.7. Determine the probability that X is between a and b. Verify that the probability model for X has the graph given in Figure 5.5.

- 2.
- For the second example above, find the probability that X is between 0 and 1/2.
- 3.
- For the second example above, find the probability that X is between 3/4 and 1.
- 4.
- For the second example above, find the probability that X is between 0 and 1/4.
- 5.
- For the second example above, (choose a point at random inside the unit circle), find
*c*so that the probability that X is between 0 and*c*is 1/2. (Hint Sketch it!).