4.3: Exponential Distributions
A third type of probability distribution is an exponential distribution. When we discussed normal distributions, or standard distributions, we talked about the fact that these distributions used continuous data, so you could use standard distributions when talking about heights, ages, lengths, temperatures, and the like. The same types of data are used when discussing exponential distributions. Exponential distributions, contrary to standard distributions, deal more with rates or changes over time. For example, the length of time the battery in your car will last is an exponential distribution. The length of time is a continuous random variable. A continuous random variable is one that can form an infinite number of groupings. So time, for example, can be broken down into hours, minutes, seconds, milliseconds, and so on. Another example of an exponential distribution is the lifetime of a computer part. Different computer parts have different life spans, depending on their use (and abuse). The rate of decay of the computer part determines the shape of the exponential distribution.
Let’s look at the differences between the normal distribution curve, a binomial distribution histogram, and an exponential distribution. List some of the similarities and differences that you see in the figures below.
Notice that with the standard distribution and the exponential distribution curves, the data represents continuous variables. The data in the binomial distribution histogram, on the other hand, is discrete. Also, the curve for the standard distribution is symmetrical about the mean. In other words, if you draw a horizontal line through the center of the curve, the 2 halves of the standard distribution curve would be mirror images of each other. This symmetry does not exist for the exponential distribution curve (nor for the binomial distribution). Did you notice anything else?
Let’s look at some examples where the resulting graphs would show you an exponential distribution.
Example 15
ABC Computer Company is doing a quality control check on their newest core chip. They randomly chose 25 chips from a batch of 200 to test and examined them to see how long they would continuously run before failing. The following results were obtained:
Number of Chips | Hours to Failure |
---|---|
8 | 1,000 |
6 | 2,000 |
4 | 3,000 |
3 | 4,000 |
2 | 5,000 |
2 | 6,000 |
What kind of data is represented in the table?
Solution:
In order to solve this problem, you need to graph it to see what it looks like. You can use graph paper or your calculator. Entering the data into the TI-84 involves the following keystrokes. There are a number of them, because you have to enter the data into L1 and L2, and then plot the lists using STAT PLOT.
After you press \begin{align*}\boxed{\text{GRAPH}}\end{align*}, you get the following curve.
This curve looks somewhat like an exponential distribution curve, but let’s test it out. You can do this on the TI-84 by pressing \begin{align*}\boxed{\text{STAT}}\end{align*}, going to the CALC menu, pressing \begin{align*}\boxed{\text{0}}\end{align*}, and pressing \begin{align*}\boxed{\text{ENTER}}\end{align*}.
Notice that the \begin{align*}r^2\end{align*} value is close to 1. This value indicates that an exponential curve is a good fit for this data and that the data, therefore, represents an exponential distribution.
You use regression to determine a rule that best explains the data you are observing. There is a standard quantitative measure of this best fit, known as the coefficient of determination (\begin{align*}r^2\end{align*}). The value of \begin{align*}r^2\end{align*} can be from 0 to 1, and the closer the value is to 1, the better the fit. In our data above, the \begin{align*}r^2\end{align*} value is 0.9856 for the exponential regression. If we had done a quadratic regression instead of an exponential regression, our \begin{align*}r^2\end{align*} value would have been 0.9622. The data is not linear, but if we thought it might be, the \begin{align*}r^2\end{align*} value would have been 0.9161. Remember, the higher the \begin{align*}r^2\end{align*} value, the better the fit.
You can even go a step further and graph the exponential regression curve on top of our plotted points. Follow the keystrokes below and test it out.
Note: It was not indicated that the data was in L1 and L2 when finding the exponential regression. This is because it is the default of the calculator. If you had used L2 and L3, you would have had to add this to your keystrokes.
Take a look at the formula that you used with the exponential regression calculation (ExpReg) above. The general formula was \begin{align*}y= ab^x\end{align*}. This is the characteristic formula for an exponential distribution curve. Siméon Poisson was one of the first to study exponential distributions with his work in applied mathematics. The Poisson distribution, as it is known, is a form of an exponential distribution. He received little credit for his discovery during his lifetime, as it only found application in the early part of the \begin{align*}20^{\text{th}}\end{align*} century, almost 70 years after Poisson had died. To read more about Siméon Poisson, go to http://en.wikipedia.org/wiki/Sim%C3%A9on_Denis_Poisson.
Example 16
Radioactive substances are measured using a Geiger-Müller counter (or a Geiger counter for short). Robert was working in his lab measuring the count rate of a radioactive particle. He obtained the following data:
Time (hr) | Count (atoms) |
---|---|
15 | 544 |
12 | 272 |
9 | 136 |
6 | 68 |
3 | 34 |
1 | 17 |
Is this data representative of an exponential distribution? If so, find the equation. What would be the count at 7.5 hours?
Solution:
Remember, we can plot this data using pencil and paper, or we can use a graphing calculator. We will use a graphing calculator here.
The resulting graph appears as follows:
At a glance, it does look like an exponential curve, but we really have to take a closer look by doing the exponential regression.
In the analysis of the exponential regression, we see that the \begin{align*}r^2\end{align*} value is close to 1, and, therefore, the curve is indeed an exponential curve. We should go a step further and graph this exponential equation onto our coordinate grid and see how close a match it is.
It is a very good match, so the equation representing our data is, therefore, \begin{align*}y = 15.06(1.274^x)\end{align*}.
The last part of our problem asked us to determine what the count was after 7.5 hours. In other words, what is \begin{align*}y\end{align*} when \begin{align*}x = 7.5\end{align*}? This question can be answered as shown below:
\begin{align*}y & = 15.06 \left (1.274^x \right )\\ y & = 15.06 \left (1.274^{7.5} \right )\\ y & = 15.06 \left (6.149 \right )\\ y & = 92.6 \ \text{atoms}\end{align*}
We can check this on our calculator as follows:
Our calculation is a bit over, because we rounded the values for \begin{align*}a\end{align*} and \begin{align*}b\end{align*} in the equation \begin{align*}y = ab^x\end{align*}, whereas the calculator did not.
Example 17
Jack believes that the concentration of gold decreases exponentially as you move further and further away from the main body of ore. He collects the following data to test out his theory:
Distance (m) | Concentration (g/t) |
---|---|
0 | 320 |
400 | 80 |
800 | 20 |
1,200 | 5 |
1,600 | 1.25 |
2,000 | 0.32 |
Is this data representative of an exponential distribution? If so, find the equation. What is the concentration at 1,000 m?
Solution:
Again, we can plot this data using pencil and paper, or we can use a graphing calculator. As with Example 16, we will use a graphing calculator here.
The resulting graph appears as follows:
At a glance, it does look like an exponential curve, but we really have to take a closer look by doing the exponential regression.
In the analysis of the exponential regression, we see that the \begin{align*}r^2\end{align*} value is close to 1, and, therefore, the curve is indeed an exponential curve. We will go a step further and graph this exponential equation onto our coordinate grid and see how close a match it is.
It is a very good match, so the equation representing our data is, therefore, \begin{align*}y = 318.56 \left (0.9965^x \right )\end{align*}.
The problem asks, “What is the concentration at 1,000 m?” This question can be answered as shown below:
\begin{align*}y & = 318.56 \left (0.9965^x \right )\\ y & = 318.56 \left (0.9965^{1000} \right )\\ y & = 318.56(0.03001)\\ y & = 9.56 \ \text{g/t}\end{align*}
Therefore, the concentration of gold is 9.56 grams of gold per ton of rock.
We can check this on our calculator as follows:
Our calculation is a bit under, because we rounded the values for \begin{align*}a\end{align*} and \begin{align*}b\end{align*} in the equation \begin{align*}y = ab^x\end{align*}, whereas the calculator did not.
Points to Consider
- Why is a normal distribution considered to be a continuous probability distribution, whereas a binomial distribution is considered to be a discrete probability distribution?
- How can you tell if a curve is truly an exponential distribution curve?
Vocabulary
- Binomial distribution
- A probability distribution of the successful trials of a binomial experiment.
- Binomial experiments
- Experiments that include only 2 choices, with distributions that involve a discrete number of trials of these 2 possible outcomes.
- Coefficient of determination \begin{align*}(r^2)\end{align*}
- A standard quantitative measure of best fit. Has values from 0 to 1, and the closer the value is to 1, the better the fit.
- Continuous data
- Data where an infinite number of values exist between any 2 other values. Data points are joined on a graph.
- Continuous random variable
- A variable that can form an infinite number of groupings.
- Continuous variables
- Variables that take on any value within the limits of the variable.
- Discrete values
- Data where a finite number of values exist between any 2 other values. Data points are not joined on a graph.
- Distribution
- The description of the possible values of a random variable and the possible occurrences of these values.
- Exponential distribution
- A probability distribution showing a relation in the form \begin{align*}y = ab^x\end{align*}.
- Normal distribution curve
- A symmetrical curve that shows the highest frequency in the center (i.e., at the mean of the values in the distribution) with an identical curve on either side of that center.
- Standard distributions
- Normal distributions, which are often referred to as bell curves.