The exponential distribution has some specific values in statistics. For example, if its rate is A then the expectation is 1/A. A generator generates electricity for a system with energy storage. If the generator fails, the system runs out of energy exactly five days later. Let the time to failure X for the generator be exponentially distributed with expectation value of 600 days while Y denotes the time until the system runs out of energy (five days later). 1. What is the probability P that the generator will fail within the first 600 days? 2. What is the quantile of the random variable Y with this probability? 3. Calculate the coefficients of variation for X and Y.

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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!