4.1: Standard Distributions
Learning Objectives
- Be familiar with the standard distributions (normal, binomial, and exponential).
- Use standard distributions to solve for events in problems in which the distribution belongs to those families.
Say you were buying a new bicycle for going back and forth to school. You want to buy something that lasts a long time and something with parts that will also last a long time. You research on the internet and find one brand “Buy Me Bike” that shows the following graph with all of its advertising.
(a) What type of probability distribution is being represented by this graph?
(b) Is the data represented continuous or discrete? How can you tell?
(c) Does the data in the graph indicate that the company produces bicycles that have a respectable life span? Explain.
Work through the lesson and then revisit this problem to determine the solution.
Now that we know a little about probability and variables, let’s move into the concept of distribution. A distribution is simply the description of the possible values of the random variables and the possible occurrences of these. For our discussions, we will say it is the probability of the occurrences. The main form of probability distribution is standard distribution. Standard distribution is a normal distribution and often people refer to it as a bell curve.
If you were to toss a fair coin 100 times, you would expect the coin to land on tails close to 50 times and heads 50 times. However, tails may not appear as expected. Look at the histograms below.
Notice that when we actually flipped the 100 coins in our experiment, we saw that tails come up 70 times and heads only 30 times. The theoretical probability is what we would expect to happen. In a regular fair coin toss, we have an equal chance of getting a head or a tail. Therefore, if we flip a coin 100 times we would expect to see 50 heads and 50 tails. When we actually flip 100 coins, we actually saw 70 tails and 30 heads. If we were to repeat this experiment, we might see 60 tails and 40 heads.
If we were to keep doing this flipping experiment, say 500 times, we may see the values get closer to the theoretical probability (the histogram on the left). As the number of data values increase, the graph of the results starts to look a bell-shaped curve. This type of distribution of data is normal or standard distribution. The distribution of the data values is shown in this curve. The more data points, the more we see the bell shape.
Between the two red lines represents 68% of the data. Between the two purple lines represents 95% of the data. Between the two blue lines represents 99.7 % of the data. You will learn more about the normal distribution in Chapter 5.
What is interesting about our flipping coin example is that it is a binomial experiment. What is meant by this is that it does not have a standard distribution but a binomial distribution. Why? This is because binomial experiments only have two outcomes. Think about it. If we flip a coin, choose between true or false, choose between a Mac or a PC computer, or even asked for tea or coffee at a restaurant, these are all options that involve either one choice or another. These are all experiments that are designed where the possible outcomes are either one or the other. Binomial experiments are experiments that involve only two choices and their distributions involve a discrete number of trials of these two possible outcomes. Therefore a binomial distribution is a probability distribution of the successful trials of the binomial experiments.
Technology Note
Let’s try the following on the graphing calculator. We are going to flip a coin 15 times and count the number of heads. Now, remember, the probability of getting a head is 50%. We are then going to repeat this experiment 25 times. On the graphing calculator, press the following:
If we wanted to look at a histogram of the data, we could store the data into a list and have a look at it.
Press [STAT PLOT] and choose the histogram function.
But what about if we were talking about 50 repetitions? Now we would type in:
But what about if we were talking about 500 repetitions? Now we would type in:
Notice as we increase the number of repetitions, we are getting closer and closer to the normal distribution from the beginning of this chapter. For data that is actually normal distributed, the sample size can be any size. So, for example, you could collect the marks from a class of students \begin{align*}(n = 30)\end{align*} and find that these are normally distributed. For binomial distributions, the sample size tends to be much larger.
Another type of distribution is called exponential distribution. If you remember, both normal distribution and binomial distribution dealt with discrete data. Discrete variables are individualized data points such as heads or tails, marks on a test, a baby being a boy or a girl, rolls on a die, etc. Essentially, these are set numbers being an either-or choice. With exponential distributions, however, the data are considered continuous. Continuous variables have an infinite number of groupings depending on what kind of scale you use. Say, for example, you surveyed your class and asked them how long it took them to walk to school. Your scale could be in minutes, in minutes and seconds, in minutes, seconds, and fractions of a second (which may seem unreasonable if you are not an Olympic Athlete). Regardless, the time measurement itself is a continuous variable. Look at the two graphs below just to see the difference between a graph of a discrete variable and the graph of a continuous variable.
For exponential distributions, the continuous data graph would change to look more like the following:
Notice, the exponential distribution curve is also showing continuous data but the graph is curved and not straight. Therefore, an exponential distribution is a probability distribution showing the relation in the form \begin{align*}y = a^x\end{align*} where \begin{align*}a\end{align*} is any positive number.
Let’s look at our example from the start of the chapter.
Say you were buying a new bicycle for going back and forth to school. You want to buy something that lasts a long time and something with parts that will also last a long time. You research on the internet and find one brand “Buy Me Bike” that shows the following graph with all of its advertising.
(a) What type of probability distribution is being represented by this graph?
(b) Is the data represented continuous or discrete? How can you tell?
(c) Does the data in the graph indicate that the company produces bicycles that have a respectable life span? Explain.
Solution:
(a) The distribution in this graph is exponential because it is a curved plot of data.
(b) The data is continuous because the data points are joined together. Discrete data points would not be joined together.
(c) In the graph, the parts will last for many years before breaking down. At 20 years, for example, the age of the parts is still equals 0.15 years.
Lesson Overview
The standard normal distribution is a normal distribution where the area under each curve is the same. When a sample is examined, and the frequency distribution is seen as normal, the resulting data displayed in a histogram often approximates a bell curve. Binomial experiments are probability experiments that would satisfy the following four requirements:
- Each trial can have only two outcomes or outcomes that can be reduced to two outcomes. These outcomes can be considered as either success or failure.
- There must be a fixed number of trials.
- The outcomes of each trial must be independent of each other.
- The probability of a success must remain the same for each trial.
The distribution curves for binomial distribution experiments appear to be normal only when the sample size increases. An exponential distribution occurs when data is continuous and in the form of \begin{align*}y = a^x\end{align*}. The resulting graphs that form are exponential curves rather than in the form of a histogram or a normal distribution curve.
Points to Consider
- How large a sample size is necessary for a binomial distribution to appear normal?
- When is exponential distribution an important distribution to use?
Vocabulary
- Standard Distribution
- A normal distribution and often people refer to it as a bell curve.
- Normal Distribution Curve
- A symmetrical curve that shows that the highest frequency in the center (i.e., at the mean of the values in the distribution) with an equal curve on either side of that center.
- Normal Distribution
- A family of distributions that have the same general shape (curve).
- Binomial Experiments
- Experiments that involve only two choices and their distributions involve a discrete number of trials of these two possible outcomes.
- Binomial Distribution
- A probability distribution of the successful trials of the binomial experiments.
- Continuous Data
- An infinite number of values exist between any two other values in the table of values or on the graph. Data points are joined.
- Discrete Data
- A finite number of data points exist between any two other values. Data points are not joined.
- Exponential Distribution
- A probability distribution showing the relation in the form \begin{align*}y = a^x\end{align*} where \begin{align*}a\end{align*} is any positive number.
Review Questions
- Is the following graph representing a normal distribution, and exponential distribution, or a binomial distribution? How can you tell?
- Is the following graph representing a normal distribution, and exponential distribution, or a binomial distribution? How can you tell?
- Is the following graph representing a normal distribution, and exponential distribution, or a binomial distribution? How can you tell?
- Is the following graph representing a normal distribution, an exponential distribution, or a binomial distribution? How can you tell?
- Is the following graph representing a normal distribution, an exponential distribution, or a binomial distribution? How can you tell?
- Is the following graph representing a normal distribution, and exponential distribution, or a binomial distribution? How can you tell?
- Describe in your own words the difference between the binomial distribution and the normal distribution.
- Find two examples of data that can be collected resulting in an exponential distribution.
Review Answers
- This is binomial since the data shows discrete frequencies and is not in the shape of a normal curve.
- This is exponential since the data shows continuous frequencies is in the shape of an exponential curve. It could represent a decay curve.
- This curve is clearly a normal distribution because it is a normal curve with an equal spread of the data on either side of the center point.
- Although this histogram is getting close to the graph of a normal distribution, it is still not equal area on either side of the mean (center point).
- This is exponential since the data shows continuous frequencies is in the shape of an exponential curve. It could represent a growth curve.
- Although this histogram is getting closer to the graph of a normal distribution, it is still not equal area on either side of the mean (center point). One could probably argue that it is both but would have to wait until a later chapter to actually learn to calculate the values of mean and standard deviation in order to prove.
- Answers will vary
- Answers will vary but speed and time are two.
Answer Key for Review Questions (even numbers)
2. This is exponential since the data shows continuous frequencies is in the shape of an exponential curve. It could represent a decay curve.
4. Although this histogram is getting close to the graph of a normal distribution, it is still not equal area on either side of the mean (center point).
6. Although this histogram is getting closer to the graph of a normal distribution, it is still not equal area on either side of the mean (center point). One could probably argue that it is both but would have to wait until a later chapter to actually learn to calculate the values of mean and standard deviation in order to prove.
8. Answers will vary but speed and time are two.