STATPLOT: Geometric Histogram#
Introduction#
In a previous section (TODO: link), we introduced the Geometric Distribution. We took a look at the geometPDF function, the probability density function, and the geometCDF, the cumulative distribution function on our TI-83/84 family of calculator. These functions give us quick ways of calculating probabilites for a Geometric Random Variable.
Recall a Geometric Random Variable counts the number of binary trials until a success occurs, where a success occurs in a single trial with probability and a failure occurs in a single trial with probability. The probability density function for a Geometric Random Variable is given by,
The domain of this function is defined on all integer values greater than or equal to 1, i.e. \(x=1,2,3,...\). This means the there is non-zero probability for all values of x greater than 1. However, the Geometric PDF still represents the probability distribution of a random variable, and for this reason, the sum of probabilities for \(x=1,2,3,...\) cannot exceed 1. Therefore, we expect the probability of x assuming a particular value should go to 0 as the value of x goes to infinity.
Activity#
Let us verify this is the case by plotting a histogram of the Geometric Distribution for the cases where \(p = 0.25, 0.375, 0.50\). In order to do this, we will need to generate a list that represents the domain of a Geometric Random Variable. As we just mentioned, the domain of a Geometric Random Variable is infinite, so we will approximate its domain with a suitably large list of values.
Create a sequence of the first 50 natural numbers starting at 1 and store the result in \(L_1\) . .. topic:: Sequence Editor
To insert a sequence into \(L_1\), type in the following commands into a TI-83/84 calculator.
\(\text{BUTTON}: \text{STAT}\)
\(\text{MENU}: \text{EDIT}\)
\(\text{1}: \text{EDIT}\)
This will bring up the List Editor. Use the arrow keys to navigate to the formula bar and press ENTER to start typing a formula,
\(\text{BUTTON}: \text{2ND}\)
\(\text{BUTTON}: \text{LIST}\)
\(\text{MENU}: \text{OPS}\)
\(\text{5}: \text{seq}\)
(insert picture of sequence editor)
Question #1
Compute the sum of the first 50 natural numbers.
Hint
Use the sum function!
Excellent. This list will represent the (truncated) domain of the Geometric Random Variable. Let’s start with \(p = 0.25\). We need to compute the value of the Geometric PDF for every element of the list we just generated.
Go to STAT > EDIT and select the formula bar for . Go to 2ND > DISTR > E: GEOMETPDF to bring up the Geometric Probability Density Function editor. Pass in the following arguments,
Question #2
What is the mean (expected value) of the Geometric Distribution when \(p=0.25\)? Round to three decimal spots.
What is the median of the Geometric Distribution when \(p=0.25\)? Round to three decimal places.
Create a relative frequency histogram using \(L_1\) as your XLIST and \(L_2\) as your FREQ.
Hint
Ensure you have a viewing WINDOW set to,
XMIN: 0
XMAX: 25
XSCL: 1
YMIN: 0
YMAX: 0.5
YSCL: 1
Question #3
Write a few sentences describing the distribution. Be sure to include descriptions of shape, center and variability.
Use the technique just described to generate a new list in \(L_3\) that represents the Geometric Distribution with \(p=0.375\). Then, generate a second new list in \(L_4\) that represents the Geoemtric Distribution with \(p=0.50\).
Question #4
What is the expected value of the Geometric Distribution when \(p=0.375\)? Round to three decimal places.
What is the expected value of the Geometric Distribution when \(p=0.5\)?
Create histograms for all three Geometric Distributions stored in \(L_2, L_3\) and \(L_4\).
Question #5
Compare and contrast the distributions when \(p=0.25, 0.375, 0.50\). What happens to the Geometric Distribution as the parameter p gets larger? Explain what this means in terms of the Geometric Random Variable.
Solutions#
TODO: jquery these into hidden elements.
1: 1275
2a: 4
2b: 3
4a: 2.667
4b: 2