Continuous Random Variables

21 Continuous Probability Functions



[latexpage]

We begin by defining a continuous probability density function. We use the function notation f(x). Intermediate algebra may have been your first formal introduction to functions. In the study of probability, the functions we study are special. We define the function f(x) so that the area between it and the x-axis is equal to a probability. Since the maximum probability is one, the maximum area is also one. For continuous probability distributions, PROBABILITY = AREA.

Example 5.1

Consider the function $f(x)=\frac{1}{20}$ for 0 ≤ x ≤ 20. x = a real number. The graph of $f(x) = \frac{1}{20}$ is a horizontal line. However, since 0 ≤ x ≤ 20, f(x) is restricted to the portion between x = 0 and x = 20, inclusive.


This shows the graph of the function f(x) = 1/20. A horiztonal line ranges from the point (0, 1/20) to the point (20, 1/20). A vertical line extends from the x-axis to the end of the line at point (20, 1/20) creating a rectangle.
Figure 5.5

$f(x)=\frac{1}{20}$  for 0 ≤ x ≤ 20.

The graph of $f(x)=\frac{1}{20}$ is a horizontal line segment when 0 ≤ x ≤ 20.

The area between $f(x)=\frac{1}{20}$ where 0 ≤ x ≤ 20 and the x-axis is the area of a rectangle with base = 20 and height = $\frac{1}{20}$.

$$\text{AREA} = 20 \left( \frac{1}{20}\right) = 1$$

Suppose we want to find the area between $f(x)=\frac{1}{20}$ and the x-axis where 0 < x < 2.


This shows the graph of the function f(x) = 1/20. A horiztonal line ranges from the point (0, 1/20) to the point (20, 1/20). A vertical line extends from the x-axis to the end of the line at point (20, 1/20) creating a rectangle. A region is shaded inside the rectangle from x = 0 to x = 2.
Figure 5.6
$$\text{AREA} = (2-0)\left( \frac{1}{20}\right) = 0.1$$
$$(2-0) = 2 = \text{base of a rectangle}$$

Reminder

area of a rectangle = (base)(height).

The area corresponds to a probability. The probability that x is between zero and two is 0.1, which can be written mathematically as P(0 < x < 2) = P(x < 2) = 0.1.

Suppose we want to find the area between $f(x)=\frac{1}{20}$ and the x-axis where 4 < x < 15.


This shows the graph of the function f(x) = 1/20. A horiztonal line ranges from the point (0, 1/20) to the point (20, 1/20). A vertical line extends from the x-axis to the end of the line at point (20, 1/20) creating a rectangle. A region is shaded inside the rectangle from x = 4 to x = 15.
Figure 5.7
$$\text{AREA} = (15-4)\left( \frac{1}{20}\right) = 0.55$$
$$(15-4) = 11 = \text{base of a rectangle}$$

The area corresponds to the probability P(4 < x < 15) = 0.55.

Suppose we want to find P(x = 15). On an x-y graph, x = 15 is a vertical line. A vertical line has no width (or zero width). Therefore, P(x = 15) = (base)(height) = $(0)\left(\frac{1}{20}\right) = 0$


This shows the graph of the function f(x) = 1/20. A horiztonal line ranges from the point (0, 1/20) to the point (20, 1/20). A vertical line extends from the x-axis to the end of the line at point (20, 1/20) creating a rectangle. A vertical line extends from the horizontal axis to the graph at x = 15.
Figure 5.8

P(X ≤ x), which can also be written as P(X < x) for continuous distributions, is called the cumulative distribution function or CDF. Notice the “less than or equal to” symbol. We can also use the CDF to calculate P(X > x). The CDF gives “area to the left” and P(X > x) gives “area to the right.” We calculate P(X > x) for continuous distributions as follows: P(X > x) = 1 – P (X < x).


This shows the graph of the function f(x) = 1/20. A horiztonal line ranges from the point (0, 1/20) to the point (20, 1/20). A vertical line extends from the x-axis to the end of the line at point (20, 1/20) creating a rectangle. The area to the left of a value, x, is shaded.
Figure 5.9

Label the graph with f(x) and x. Scale the x and y axes with the maximum x and y values. $f(x) =\frac{1}{20}$, 0 ≤ x ≤ 20.

To calculate the probability that x is between two values, look at the following graph. Shade the region between x = 2.3 and x = 12.7. Then calculate the shaded area of a rectangle.

This shows the graph of the function f(x) = 1/20. A horiztonal line ranges from the point (0, 1/20) to the point (20, 1/20). A vertical line extends from the x-axis to the end of the line at point (20, 1/20) creating a rectangle. A region is shaded inside the rectangle from x = 2.3 to x = 12.7
Figure 5.10
$P(2.3<x<12.7) = \text{(base)(height)} = (12.7-2.3)\left(\frac{1}{20} \right) = 0.52$

Try It 5.1

Consider the function $f(x)=\frac18$ for 0 ≤ x ≤ 8. Draw the graph of f(x) and find P(2.5 < x < 7.5).

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