Now that we know a little about Bernoulli trials and the binomial distribution, let's take a more visual look at the distribution using technology.
The graph of the binomial distribution
n is the number of trials of a Bernoulli experiment (an experiment with only two outcomes, a success or a failure)
p is the probability of success of each trial and each trial is independent.
We looked at graphs of general discrete random variables earlier, but here we will specifically explore the shape and distribution of binomial random variable graphs using technology. We will draw and generate the probability histogram for a given value of n and p, exploring the general shape and then the effect of varying n and p.
Exploring the effect of p
Using the Geogebra applet below, examine the graph of X~B\left(15,0.5\right)
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How would you describe the distribution of the graph you see?
Remember when describing the shape of a histogram we use the phrases positively skewed, symmetrical and negatively skewed.
We will now examine the graphs of X~B\left(15,p\right), changing the values of p. For each, draw a sketch of the graph and describe the distribution you see.
| p | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 |
|---|---|---|---|---|---|---|---|---|---|
| Graph | |||||||||
| Shape |
Write a sentence to explain what you notice about the shape of the distribution as p varies.
Exploring the effect of n
Let's return to the graph of X~B\left(15,0.5\right)
We will now vary n, and explore various graphs of the form X~B\left(n,0.5\right)
Use the Geogebra applet below to explore what happens when we change n.
Set p=0.5 and n=15.
Increase n in increments of 1, sliding up to n=30.
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What do you notice happens to the shape of the graph? Does the skewness change? If not, what changes?
Now decrease the value of n towards 0. What do you notice happens to the shape of the distribution?
Summarise your findings in one sentence.
Consolidation questions
- Using your findings above, describe the difference between the graph of X~B\left(20,0.2\right) and Y~B\left(20,0.85\right)
- Using your findings above, describe the difference between the graph of X~B\left(5,0.3\right) and Y~B\left(50,0.3\right)
Binomial graphs, measures of centre and spread
When we combine our knowledge about the distribution of a histogram and its influence on the mean and standard deviation of a data set, what can we deduce about the graph of a binomial distribution, and the influence of p and n on the expected value and the standard deviation of the distribution?
The mean or expected value of X, a binomial random variable, is calculated by:
E(X)=np
The variance of X is calculated by:
Var(X)=np(1-p)
And thus the standard deviation of X is calculated by:
\sigma(X)=\sqrt{np(1-p)}
Exploration tasks
Task 1
Consider X~B\left(20,p\right)
We will vary p to explore what happens to the mean and standard deviation of the distribution for various levels of skewness. Let's tabulate our findings as follows:
| p |
Positively skewed p=0.1 |
Symmetrical p=0.5 |
Negatively skewed p=0.85 |
|---|---|---|---|
| Graph | |||
| E(X) | |||
| StDev(X) |
Task 2
By considering what you observed above, try answering the following questions without doing any calculations and then confirm you were correct with the relevant calculations.
- Which of the following distributions would have the higher mean? X~B\left(20,0.1\right) or Y~B\left(20,0.5\right). Explain your reasoning.
- Which of the following distributions would have the lower standard deviation? X~B\left(20,0.55\right) or Y~B\left(20,0.87\right). Explain your reasoning.
- Which of the following distributions would have the lower mean? X~B\left(20,0.25\right) or Y~B\left(20,0.76\right). Explain your reasoning.
- Which of the following distributions would have the higher standard deviation? X~B\left(20,0.35\right) or Y~B\left(20,0.8\right). Explain your reasoning.
Task 3
Let's summarise our findings by completing the following statements: (you might like to use the words in this box)
| increases | decreases | smaller | larger | positively |
| negatively | symmetrical | skewed | maximum | minimum |
The mean and varying p
For a binomial random variable with the number of trials n, as p decreases towards 0 the mean __________. Algebraically this occurs because n is being multiplied by an increasingly _________ number. Graphically this occurs because the graph is becoming more ___________ skewed.
As p increases towards 1 the mean __________. Algebraically this occurs because n is being multiplied by an increasingly __________ number. Graphically this occurs because the graph is becoming more _________ skewed.
The standard deviation and varying p
For a binomial random variable with a number of trials n, and p=0.5, we have a __________ graph and the standard deviation is at its ___________ value.
As p decreases towards 0 the graph becomes more ___________ and thus the standard deviation _________. Algebraically we are finding the square root of an increasingly __________ value and this confirms our result.
Similarly, as p increases towards 1 the graph becomes more ___________ and thus the standard deviation __________. Algebraically we are finding the square root of an increasingly __________ value and this confirms our result.