Example 11: alarm system thresholds Consider a burglar alarm system which uses motion detector (movement above a threshold will trigger an alarm). Random motion has a Rayleigh pdf with variance equal to =1.5V². Testing shows an intruder causes a random motion with voltage output vintruder that has a Gaussian distribution ~N(7.5,4). If the threshold level is set low enough, the random motion can reach a level high enough to trigger a false alarm. a) Determine the threshold level such that the probability of false alarm PEA is under 0.005. b) Calculate the resulting probability of detecting an intruder Pp. c) Compute the motion detector threshold level needed to ensure Pp >0.99. Compute the resulting PFA:
Continuous Probability Distributions
Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.
Normal Distribution
Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!
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