SOCR EduMaterials Activities ExpCarTimeExperiment

Description

TBD.

Goal

To provide an interactive simulation demonstrating exponential processes, exponential decay, simulation and model fitting. Additional aims are to:

• Draw analogy between Gamma, Exponential and Poisson (discrete!)
• Provide the means for (large) data generation/sampling
• Provide the infrastructure for model-fitting and assessment.

Experiment

Go to the SOCR Experiments and select the Exponential Car-Time Experiment from the drop-down list of experiments on the top left. The image below shows the initial view of this experiment with pointers and highlights of the most important components and controls:

Select the number of cars (k) and the rate of the exponential process (r). When pressing the play button, one trial will be executed that involves sampling k random exponential time variables (representing the between car times). The results will be displayed graphically in the two graph windows (timeline and histogram) for this process. Summary statistics and model vs. data distributions are shown in the distribution table below. The stop button ceases any activity and is helpful when the experimenter chooses a large number of cars (k is large). The reset button will clear and re-initialize the entire experiment, deleting all previous information and data collected.

As the number of cars (k) increases, the distribution of the between-car times will have exponential-decay shape. Note that as the number of cars (k) increases, the empirical histogram (graph) and moments (table) of the random exponential times begin to converge to their corresponding Exponential distribution counterparts. Arrows point to the match of the empirical histogram of the between-car times and theoretical Exponential distribution, the random sample of between-car times (Y column of left table) and the similarity between the first two moments of the sample data and model (Exponential) distributions.

Applications

Analogy between Gamma, Exponential and Poisson distributions

Number of Arrivals in a fixed time-period follows Poisson distribution.

Exponential data sampling

We can use this applet to geenrate a sample of an arbitrally size from an Exponential Distribution with specified rate parameter. Then these data may be used for simulations or exploratory data analsys. For example, suppose we ran the experiment with k=1,000, as shown onthe image above. We can copy the sample from the Y column and past it in any of the SOCR Charts. The image below shows the construction of a Statistical Bar Chart using the 1,000 Exponential (between-car) time measurements. Notice how the sumamry statistics on this Bar Chart coinside with the data-statistics reported on the distribution table of the Exponential Time Car Experiment applet.

Exponential model-fitting and assessment

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