4.4 Continuous Random Variables

Continuous random variables (CRVs) have many applications. Baseball batting averages, IQ scores, the length of time a long distance telephone call lasts, the amount of money a person carries, the lifespan of a computer chip, and SAT scores are just a few. The field of reliability depends on a variety of continuous random variables.

NOTE: The values of discrete and continuous random variables can be ambiguous. For example, if X is equal to the number of miles (to the nearest mile) you drive to work, then X is a discrete random variable. You count the miles. If X is the distance you drive to work, then you measure values of X, which is a continuous random variable. For a second example, if X is equal to the number of books in a backpack, then X is a discrete random variable. If X is the weight of a book, then X is a continuous random variable because weights are measured. How the random variable is defined is very important.

Properties of Continuous Probability Distributions

The graph of a continuous probability distribution is a curve. Probability is represented by the area under the curve.

The curve—represented by the symbol f(x)—is called the probability density function (PDF). f(x) is the function that corresponds to the graph; we use the density function f(x) to draw the graph of the probability distribution.

Area under the curve is given by a different function called the cumulative distribution function (CDF). The cumulative distribution function is used to evaluate probability as area.

When dealing with CDFs:

  • The outcomes are measured, not counted.
  • The entire area under the curve and above the x-axis is equal to one.
  • Probability is found for intervals of x values rather than for individual x values.
  • P(c < x < d) is the probability that the random variable X is in the interval between the values c and d. P(c < x < d) is the area under the curve, above the x-axis, to the right of c, and the left of d.
  • The probability that x takes on any single individual value is zero: P(x = c) = 0. The area below the curve, above the x-axis, and between x = c and x = c has no width and therefore no area (area = 0). Since the probability is equal to the area, the probability is also zero.
  • P(c < x < d) is the same as P(c ≤ x ≤ d) because probability is equal to area.

We will find the area that represents probability by using geometry, formulas, technology, or probability tables. In general, calculus is needed to find the area under the curve for many probability density functions; however, much of the work has already been done for us. The formulas to find the area in this textbook have already been found by using the techniques of integral calculus.

Some Continuous Distributions

There are many continuous probability distributions. When using a continuous probability distribution to model probability, the distribution used is selected to model and fit the particular situation in the best way. We do not often handle general CRVs, instead studying special known cases most of the time. The following graphs illustrate some of these these distributions.

Figure description available at the end of the section.
Figure 4.11: Continuous distributions. Figure description available at the end of the section.

Probability Density Functions

We begin by defining a continuous probability density function, using the function notation f(x). 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, you can think about it as: PROBABILITY = AREA.

The Uniform Distribution

The (continuous) uniform distribution is fairly simple and is a great place to start in demonstrating the ideas of continuous distributions. It is concerned with events that are equally likely to occur. When working out problems that have a uniform distribution, be careful to note if the data is inclusive or exclusive of endpoints.

The notation for the uniform distribution is:

X ~ U(a, b), where a = the lowest value of x and b = the highest value of x.

The probability density function is:

f(x) = \frac{1}{b-a} for a ≤ x ≤ b.

Formulas for the theoretical mean and standard deviation are:

\mu =\frac{a+b}{2} and \sigma =\sqrt{\frac{{\left(b-a\right)}^{2}}{12}}

Example

Suppose a researcher tracked the duration each time his eight-week-old baby smiled. The following data are 55 smiling times, in seconds.

Smiling times (in seconds)
10.4 19.6 18.8 13.9 17.8 16.8 21.6 17.9 12.5 11.1 4.9
12.8 14.8 22.8 20.0 15.9 16.3 13.4 17.1 14.5 19.0 22.8
1.3 0.7 8.9 11.9 10.9 7.3 5.9 3.7 17.9 19.2 9.8
5.8 6.9 2.6 5.8 21.7 11.8 3.4 2.1 4.5 6.3 10.7
8.9 9.4 9.4 7.6 10.0 3.3 6.7 7.8 11.6 13.8 18.6

Figure 4.12: Smiling times

The sample mean is 11.49, and the sample standard deviation is 6.23.

We will assume that the smiling times, in seconds, follow a uniform distribution between zero and 23 seconds, inclusive. This means that any smiling time from zero to and including 23 seconds is equally likely. The histogram that could be constructed from the sample is an empirical distribution that closely matches the theoretical uniform distribution.

Let X represent length, in seconds, of an eight-week-old baby’s smile.

For this example:

X ~ U(0, 23) and f(x) = \frac{1}{23-0} for 0 ≤ X ≤ 23.

For this problem, the theoretical mean and standard deviation are:

μ = \frac{0\text{ }+\text{ }23}{2}} = 11.50 seconds and σ = \sqrt{\frac{{\left(23\text{ }-\text{ }0\right)}^{2}}{12}}}= 6.64 seconds.

Notice that the theoretical mean and standard deviation are close to the sample mean and standard deviation in this example.

Example

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.
Figure description available at the end of the section.
Figure 4.13: Example of a function. Figure description available at the end of the section.
  • 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}.
  • Area = 20(\frac{1}{20}) =1

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

Figure description available at the end of the section.
Figure 4.14: Finding area. Figure description available at the end of the section.
Solution

Area = (2 – 0)(\frac{1}{20}) = 0.1

(2–0) = 2 = 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.

Figure description available at the end of the section.
Figure 4.15: Finding area. Figure description available at the end of the section.
Solution

Area = (15 – 4)(\frac{1}{20}) = 0.55

(15 – 4) = 11 = base of a rectangle

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

 

Suppose we want to find P(x = 15).

Figure description available at the end of the section.
Figure 4.16: Finding a value. Figure description available at the end of the section.
Solution

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)(\frac{1}{20}) = 0.

 

Recall that 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).

Figure description available at the end of the section.
Figure 4.17: Area to the left. Figure description available at the end of the section.

Label the graph with f(x) and x. Scale the x- and y-axes with the maximum x and y values.

Solution

f(x) = \frac{1}{20}, 0 ≤ x ≤ 20.

 

Figure description available at the end of the section.
Figure 4.18: Finding area. Figure description available at the end of the section.

 

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

 

Solution

𝑃(2.3<𝑥<12.7) = (base)(height) = (12.7−2.3)(\frac{1}{20}) = 0.52

Your Turn!

Consider the function f(x) = \frac{\text{1}}{8} for 0 ≤ x ≤ 8. Draw the graph of f(x), and find P(2.5 < x < 7.5).

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Figure References

Figure 4.11: Kindred Grey (2020). Continuous distributions. CC BY-SA 4.0. Adaptation of Figures 5.37, 5.38, and 5.39 from OpenStax Introductory Statistics (2013) (CC BY 4.0). Retrieved from https://openstax.org/books/statistics/pages/5-practice

Figure 4.13: Kindred Grey (2020). Example of a function. CC BY-SA 4.0.

Figure 4.14: Kindred Grey (2020). Finding area. CC BY-SA 4.0.

Figure 4.15: Kindred Grey (2020). Finding area. CC BY-SA 4.0.

Figure 4.16: Kindred Grey (2020). Finding a value. CC BY-SA 4.0.

Figure 4.17: Kindred Grey (2020). Area to the left. CC BY-SA 4.0.

Figure 4.18: Kindred Grey (2020). Finding area. CC BY-SA 4.0.

Figure Descriptions

Figure 4.11: Three distributions side by side. Left: Bell-shaped graph; the symmetric graph reaches maximum height at x = 0 and slopes downward gradually to the x-axis on each side of the peak. Middle: downward sloping line graph; it begins at a point on the y-axis and approaches the x-axis at the right edge of the graph. Right: The horizontal axis ranges from 0 to 10. The distribution is modeled by a rectangle extending from x = 3 to x =8.

Figure 4.13: This shows the graph of the function f(x) = 1/20. A horizontal 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 4.14: This shows the graph of the function f(x) = 1/20. A horizontal 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 4.15: This shows the graph of the function f(x) = 1/20. A horizontal 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 4.16: This shows the graph of the function f(x) = 1/20. A horizontal 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 4.17: This shows the graph of the function f(x) = 1/20. A horizontal 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 4.18: This shows the graph of the function f(x) = 1/20. A horizontal 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

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