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# Exponential Distributions

## Distributing data based on an exponential rate of change.

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Exponential Distributions

You're working in a factory where batteries are made. You're responsible for testing the batteries to determine their reliability for the end consumer. How do you think you'd collect the testing data? No matter how well they're made, batteries still wear out over time. If you test 2,000 batteries, how do you think the final data regarding battery failure will look on a graph?

### Guidance

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?

The general formula for an exponential distribution curve is \begin{align*}y= ab^x\end{align*}. 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.

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.

Let’s look at some examples where the resulting graphs would show you an exponential distribution.

#### Example A

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? Test out your answer with regression.

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 0.9856, which 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.

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.

#### Example B

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?

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 C

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?

Again, we can plot this data using pencil and paper, or we can use a graphing calculator. As with Example B, 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

• How can you tell if a curve is truly an exponential distribution curve?

### Vocabulary

Continuous data is data where an infinite number of values exist between any 2 other values. With continuous data, data points are joined on a graph. A continuous random variable is a variable that can form an infinite number of groupings. Both standard distributions, which are normal distributions, or bell curves, and an exponential distribution, which is a probability distribution showing a relation in the form \begin{align*}y = ab^x\end{align*}, are graphs of continuous random variables. The coefficient of determination \begin{align*}(r^2)\end{align*} is a standard quantitative measure of best fit. It has values from 0 to 1, and the closer the value is to 1, the better the fit.

### Guided Practice

Thomas is studying for his AP Biology final. In order to complete his course, he must do a self-directed project. He decides to swab a tabletop in the student lounge and test for bacteria growing on the surface. Every hour, he looks in his Petri dish and makes an estimate of the number of bacteria present. The following results were recorded.

Time (hr) Bacteria Count
0 1
1 6
2 40
3 215
4 1,300
5 7,800

Is this data representative of an exponential distribution? If so, find the equation. What is the count after 1 day?

First we must plot the data either using pencil and paper or using a graphing calculator.

The resulting graph looks like the following.

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 is \begin{align*}y = 1.024(5.985^x)\end{align*}.

The question asks, “What is the count after 1 day?” In other words, “What is \begin{align*}y\end{align*} when \begin{align*}x = 24\end{align*}?” The answer can be calculated as follows:

\begin{align*}y & = 1.024(5.985^x)\\ y & = 1.024(5.985^{24})\\ y & = 1.024(4.46 \times 10^{18})\\ y & = 4.57 \times 10^{18} \ \text{bacteria}\end{align*}

Therefore, there are \begin{align*}4.57 \times 10^{18}\end{align*} bacteria on the tabletop after 1 day.

We can check this on our calculator as follows:

Our calculation is a bit off, because we rounded the values for a \begin{align*}a\end{align*} and \begin{align*}b\end{align*} in the equation \begin{align*}y = ab^x\end{align*}, where the calculator did not.

### Practice

1. If you watch a grasshopper jump, you will notice the following trend:
Jump Number Distance (m)
1 4
2 2
3 1.1
4 0.51
5 0.25
6 0.13

Is this data representative of an exponential distribution? If so, find the equation. Why do you think the grasshopper's distance decreased with each jump?

1. Use the equation you found in question 1 to estimate the grasshopper's distance on jump number 10.

An exponential regression produced the following results:

1. What is the exponential equation?
2. What is the coefficient of determination?
3. What is the approximate value of y when x is 5?
4. How likely is it that the exponential equation is correct?

An exponential regression produced the following results:

1. What is the exponential equation?
2. What is the coefficient of determination?
3. What is the approximate value of y when x is 8?
4. How likely is it that the exponential equation is correct?

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### Vocabulary Language: English Spanish

TermDefinition
coefficient of determination A standard quantitative measure of best fit, denoted $(r^2)$. It 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. With continuous data, data points are joined on a graph.
continuous random variable A variable that can form an infinite number of groupings. Both standard distributions and exponential distributions are graphs of continuous random variables.
standard distribution Normal distributions, or bell curves.