An astronomer wants to measure the distance from his observatory to a distant star. However, due to atmospheric disturbances, any measurement will not yield the exact distance. As a result, the astronomer has decided to make n series of measurements and then use their average value as an estimate of the actual distance. If the astronomer believes that the values of the successive measurements are independent random variables that follow approximately a normal distribution with a mean of 10 light years and a standard deviation of 2 light years, how many series of measurements he needs to make so that at least 95 percent of the mean distance is less than 9.5 light years?
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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