ECEN 390 - Laser Tag Project

By now, you have seen that many engineering problems can be modeled with random variables. For example, you learned in ECEn 240 that Ohm’s law dictates the exact voltage drop V across a resistor as a function of the current across the resistor I and the resistor’s resistance value R. That relationship is now familiar to you, and can be expressed as:

Yet, if you were to measure the resistance, the current, and the voltage, you probably wouldn’t be surprised to see a small deviation in the resulting relationship between the three measurements. In other words, you might actually observe:

for some small, but nonzero, value of epsilon.

Often these small differences occur from the inaccuracy of our measurements, and often they occur due to small differences in the environments or small defects in the materials. The combination of several of these small effects can easily be modeled by letting epsilon be a random variable with a certain distribution defined over its range of values. The Gaussian distribution tends to model several natural phenomena, resulting in its importance in statistical analysis. In many engineering applications, we want actual measurements to inform our understanding of the distribution of epsilon. Is it really Gaussian? What are the mean and variance of the distribution? Statistics can help us answer these questions.

Now, let’s apply this principle to your overall laser tag system. Although you have verified that your system “works” at 40 feet, we might naturally ask a follow on question: “How well does it work?” To answer this question, the principles of STAT 201 can help us form a meaningful statistical analysis. In other words, we will let W be a random variable with its distribution defined as:

where W = 1 indicates that your laser tag receiver system detects a ‘hit,’ and W = 0 indicates that your system misses a detection. (This type of random variable is known as a Bernoulli random variable, and you probably recognize it as one of the simplest ways to model outcomes of a random experiment.) Missed detections may occur for a number of reasons. You should brainstorm several reasons why these occur while you conduct this experiment.

The purpose of this assignment is to apply probabilistic and statistical tools from STAT 201 to help you understand the reliability of the engineering design of your laser tag system.

1. Determine the number of times n you want to test your system at each of the three distances: 20 ft, 40 ft, and 60 ft. You will be using your data to estimate probabilities of detection p_20 , p_40 , and p_60 , where the subscript indicates the distance between transmitter and receiver of your laser tag system. Justify your choice of n.
   1. In choosing a value for n, you will want to strike a balance between the accuracy of your estimates and the time it takes you to run your experiments.
   2. Remember from STAT 201 that you can develop a confidence interval for your estimate. If all of you were to run the same experiment to estimate a probability p, and then calculate confidence intervals at the 95% level of confidence for your estimate, we would expect 95% of you, on average, to actually bound the true value of p within your upper and lower bounds of the confidence interval. See Section 5.2 (pp. 338-344) of your STAT 201 textbook for a reference. Given a desired width of your confidence interval, and no knowledge of p, Example 5.14 may be useful in determining the value of n you require. Use a 95% confidence interval for your calculations.
   3. Notice that the width (difference between upper and lower bounds) of the confidence interval shrinks as n grows. This matches our intuition that more data will result in better estimates.
2. Test your laser tag system n times at each of the three distances: 20 ft, 40 ft, and 60 ft, and estimate probabilities of detection p_20 , p_40 , and p_60 . Does it make sense that these probabilities will be different? Hypothesize the relationships between these probabilities before you begin. That is, which one will be the largest, which one will be the smallest, and by about how much will they differ? How well do your estimates match your hypotheses?
   1. As you may recall from STAT 201, the maximum likelihood estimate of a probability is simply the ratio of successful trials to the number of total experiments.
3. Produce a table summarizing your results. Include the value of n, the number of successful trials at each distance, your estimates of the probabilities p_20 , p_40 , and p_60 , and 95% confidence intervals for each estimate.

Produce a short report summarizing the design and results for your experiments. Include key equations and calculations, justifications for your design choices, and your table summarizing your results. Include a list of several factors that may be leading to the randomness of your outcomes. Draw conclusions on your experiments. Finally, if you were to sell this product, how would you market the range at which your laser tag system “works?”