Next: Parameters Up: Continuous Probability Models Previous: Continuous Probability Models

# Uniform Probability Model

The probability model of a discrete random variable was evident to you before you took this course. Sure, the notation is new but in playing games like craps you knew the probabilities of interest; eg, probability of a "7". Recall that in general, the probability model of a discrete random variable with range consists of probabilities P(X = i) for i = 1,2, ..., k. The probability of a continuous random variable is not as evident. Lets begin with an example where we know the answers. This will motivate the continuous probability model.

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.
Are you ready for the big jump? What's the probability that X is between a and b when a and b are real numbers between 0 and 1? It's b - a, the length of the interval. Go back to the list above and check if this isn't so for those cases. This leads to the following, though:
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.
Note that we could continue this list forever. Each of the above intervals contains the number 1/2 and that further the length of each succeeding interval is getting smaller. Hence, we must have P(X = 1/2) = 0. But this is true for any real number a between 0 and 1; i.e., P(X = a) = 0. In general, for continuous random variables the discrete probability model will not work. But the probabilities of intervals are the probabilities of interest and this is how we define the model.

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).

Exercise 6.1.1
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!).

Next: Parameters Up: Continuous Probability Models Previous: Continuous Probability Models

2001-01-01