If the annual proportion of new restaurants that fail in a given city may be looked upon as a random vari-able having a beta distribution with α = 1 and β = 4, find (a) the mean of this distribution, that is, the annual pro-portion of new restaurants that can be expected to fail in the given city;(b) the probability that at least 25 percent of all newrestaurants will fail in the given city in any one year.
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!
able having a beta distribution with α = 1 and β = 4,
portion of new restaurants that can be expected to fail in
(b) the
restaurants will fail in the given city in any one year.
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